Let me share an argument with a gap.
Assume contrary. By rescaling the second polyhedron, say $P$, we can assume that the radius of the sphere stays the same as for the original polyhedron, say $Q$, but all edges get shorter.
Consider the central projection of the edges of $Q$ to the sphere. Note that they cut the sphere into cyclic polygons. Cyclic polygon maximize the area among all polygons with the same sides.
In general, the area is not maximal for polygons with at most the same sides --- this is the problem with my argument. For many cyclic polygons it is true --- further assume it is true for every facet of $Q$.
Now project centrally, old edges of $P$ to the sphere. We arrive at a contradiction, since the area of the sphere stays the same and all polygons are getting smaller.