So far, I'm about at the point of describing the trivial version of the problem I mentioned above. I doubt that I'll have any extra development time to get back to this until at least next year, however. Here's the comment block from a snippet of code I managed to get to:
Code:
// piecewise octahedral to equirectangular projection.
// question posed by Naima on Cartographers Guild at
// https://www.cartographersguild.com/showthread.php?t=51893 on 2023-09-20

// original input has a vertically-oriented 1:2 image (1024wx2048h) and that's
// how the solution was formulated. Probably would have been a little easier
// to require a horizontal format on the input, but it's not that bad.


/*

        |<------- W1  ------->|
        |                     |
   ---- +----------C----------+            
    ^   |\                   /|            
    |   |  \               /  |            
    |   |    \     2     /    |            
    |   |      \       /      |            
    |   |        \   /        |            |<--------------------- N --------------------->|
    |   D    3     N     1    B            +-----------+-----------+-----------+-----------+
    |   |        /   \        |            |           |           |           |           |
    |   |      /       \      |            |           |           |           |           |
    |   |    /     0     \    |            |     0     |     1     |     2     |     3     |
    |   |  /               \  |            |           |           |           |           |
        |/                   \|            |           |           |           |           |
    H1  +----------A----------+            +-----A-----+-----B-----+-----C-----+-----D-----+
        |\                   /|            |           |           |           |           |
    |   |  \               /  |            |           |           |           |           |
    |   |    \     4     /    |            |     4     |     5     |     6     |     7     |
    |   |      \       /      |            |           |           |           |           |
    |   |        \   /        |            |           |           |           |           |
    |   D    7     S     5    B            +-----------+-----------+-----------+-----------+
    |   |        /   \        |            |<--------------------- S --------------------->|
    |   |      /       \      |            
    |   |    /     6     \    |            
    |   |  /               \  |            
    v   |/                   \|            
   ---- +----------C----------+            

The idea is to take triangles on the left and warp them to squares on the right.
The points marked N and S are the north and south poles, respectively.
Line segments ABCD are the equator.
Segments B and D are measured in opposite directions (away from A)

block       source points          dest points
            w1, h1                 w2, h2
              
0           0.00, 0.50             
            1.00, 0.50             
            0.50, 0.25             
*/
I didn't quite manage to finish up the mapper points before I got pulled away, but it should be fairly simple to implement if I get an otherwise unoccupied half day or so. It doesn't require a generalized texture mapper because everything is nicely axis aligned, but it might be easier to do that way if using a tool that has that feature implemented (which Wilbur does not).