I've been using a leather belt with green compound (and bare) for final stropping of my 1095, Case CV, Opinel 'carbone' (XC90 steel) and 420HC blades. I use it like the hanging (barber) strops, and find it really does a great job in cleaning up the very fine burrs and other debris that's left
on the apex after most of the other coarser honing is done. Really makes edges on these steels pop, as a finishing step. I'm inclined to believe this is what makes the 'real' barber's strops so appealing for use with shaving razors.
For refining the
bevels of blades (behind the apex), I like hard/very firm surfaces like wood, or paper over glass, used with compound. This can also do a good job refining the apex, but I tend to think the bigger effect is on the bevels themselves. Almost without exception, I follow this with stropping on the belt as described above. To a large degree, edges can always be cleaned up a little more this way (I sometimes think of my leather belt-strop as 'the cleaner' for finishing edges).
So, to the original question of 'rigid or flexible?', I'd say BOTH.
David