(first - if you are not experienced with this and you are going to do it to a LN plane, this may not be for you. If you're worried about returning something, then call them to consult and see what they want you to do. I suppose they would advise me that I couldn't improve on their plane, but I don't know that - I'd be hesitant to suggest a buyer "fix" something by hand if I were them. If you ruin your own plane, I'm not responsible for it)
When I made my bench, I planed an area on the bench so that I could lay a 3/8" glass sheet on it and no matter where I put pressure, I couldn't fit a .0012" feeler under a 24" starrett straight edge.
That gives a sense of the flatness I'm working with. Once you have that, It's easy to start overthinking things when doing this process ("the paper will be far less flat than the lap, etc, this won't work"). Trials and proving this wrong is the answer, not quitting. I have proved it wrong.
If you are going to remove high toe and heel, no other skill is needed. If you are going to try to hit the same flatness target as shown here on a convex plane or a twisted plane (I do this only on high value infills, so that if I ever sell them, someone will get a similar feel to a premium plane with thin shavings, and not be left guessing - and if a plane is really close, I don't make it perfect). The process for a convex plane is to progressively draw file the center out of the plane until there is a tiny ring only a thousandth or two high around the outside of the plane and then lap that off. precision better than any manufactured plane is gotten pretty easily that way.
before I put new paper down on the lap each time, I use an old chisel to scrape off everything on the glass, otherwise the adhesive left behind starts to grow and a grit might stick to it. Paper is then adhered carefully (no bubbles), but not with faffery - just roll it out and stick it down - start it straight as you can't redirect the paper back toward the center of the lap if you start off line.
I use 80 grit PSA roll for *all* flattening, to improve the surface finish, see the end of this. 80 grit paper stoned with an oilstone provides fabulous flatness and no real tooth left on the sole to grab wood.
back to this one - a .0015" feeler just catches under the starrett edge right in the middle of the plane. it's not that bad, but it's problematic for fine work.
* start on the glass lap - carefully "plane" with the plane on the glass lap favoring no end over another end. About 40 or so passes, lifting the plane at the end of each to prevent swirling scratches (if you don't care about those, don't worry about it then), then check progress
(sorry, I didn't take a picture of the plane at this point). Grit and swarf needs to be wiped off of the plane sole and if any got on the straight edge, it needs to be swiped across your shirt- tiny things give misreadings. I use a starrett edge for this - when I got it, 24 inches of starrett #380 was $62. It's probably more now. It's not necessary for woodworking, maybe, but for this stuff, it is. It's nice to have for woodworking too, but mine is getting a little beat up. I check it against a 48" reference straight edge once in a while (that I don't use much for fear that I'll damage it, plus, it's heavy) and it's remained flat)
* continue on once progress is checked and the problem seen checking with a straight edge is confirmed by where the lap is removing metal. Care is taken to use the plane in the planing motion at this point to avoid dubbing edges. Be deliberate, work at 2/3rds the speed that you feel you could and lift the plane deliberately if swirls would be an issue. a few misplaced 80 grit swirls maybe hard to remove cosmetically
* as checking continues, it becomes clear that one end of the plane is approaching flatness. For me, from experience, it's usually the toe - the same is true in this case. It's OK to continue on from this point and flatten the whole plane, but the toe will end up just a tiny bit high (just enough to pass light) and more than just the last 1/2 inch or so. I don't like that. It doesn't matter, but I know how to avoid it.
* I switch to one hand on the rear handle and one hand on the rear sole of the plane applying pressure (directed pressure, sound familiar?). The entire plane sole stays in contact with the lap, careful straight strokes, but pressure is directed where metal removal is desirable. This is continued, checking every 40 or so strokes. If cosmetics are important to you and you're going no further than 80 grit, clean the paper off as you're finishing, or the areas of metal dust will leave some visual burnishing on the sole, usually in the middle. It's cosmetic, but it's there.
If necessary to avoid any more lapping of the front, turn the plane around. My bias is for the front of the plane to get lapped more, so I turn the plane around to finish directed pressure strokes and completely cease any material lapping of the front (still personal preference - the toe a thousandth or two high is no big deal, it's a favorable bias).
At some point, checking reveals no light anywhere with the starrett edge, and this stops.
On coarse grit, you can't rely on seeing scratches across the sole as the swarf and coarse paper will scratch areas a little when they're still somewhat hollow. I have an old 7 still on the shelf waiting to be addressed because I made the scratch pattern uniform. It still nips the ends off of boards a little bit. It looks nice though, but the result of the unfinished job makes it so that I don't use it because i have other planes that are flat.
here's the plane before final measurement - you can see where the swarf was building up on the paper due to burnishing (I have another step yet, so I didn't care)
back, middle and front check with the starrett edge when done (I can't get a picture in focus of it all at once).
The last step for me is to wrap 220 grit psa paper around a sandflex block or wooden block for cosmetic purposes. this is also done very carefully to keep strokes all in line so that this plane doesn't look hand lapped when done. When I sell it, I will disclose that it was. I will disclose why and what the result was, too.
This 220 grit work is cosmetic and only takes a minute. It doesn't threaten flatness, but doing a lot of lapping on 220 grit paper as a separate step on the lap might make the toe and heel proud. No like!!
I normally oilstone the sole at this point, but I figured I would just sandflex this one. The result isn't absolute cosmetic perfection, but it's good enough (had to move to the window to get an even dull light - the overhead lights cause too much glare).