Nope I was too lazy to update the cases of non-parallel "sections"¹, only mentioned it in reply to hdb (whose solution makes mine obsolete).

But I think that hdb's solution resp. your generalization already describe the complete solution room and that all possible magic square can be found there by mapping rows to diagonals.

So since the other way round works you will always find a back-projection from magic to "limbic" square.

Too tired to dig into proving it, but looking into the literature for magic squares should show it's trivial.

(unproven opinion)

update

¹) and I'm still not sure if L~R really wants them.