At the end of the original post, I said "the two solutions of which I am aware both have single digits in each blank"
I was wrong. The second single-digit "solution" I found was not in fact a solution. There are two solutions, but one is composed only of single-digit numbers while the other is not so limited.
I have read that there are only two solutions, and seeing how to find both of these leads me to be pretty certain that is correct. But a rigorous proof of that is another matter...