Do you want more or less convex?
Obvioulsy, the stiffer the material, the flatter the paper is held and the less convex the edge. You can sharpen a steep convex on a hard board, but you can't do a shallow convex if the support is too soft.
The mats that are most often used are the neoprene/cloth version, rather than the plastic faced rubber variety. With these, it doesn't matter much since the fabric does not add any stiffness to speak of.