pick a center $C$, and break up the polyhedron based on its edges, so that each edge is associated with the two triangles of its faces that it makes along with the point where a sphere with that center would meet each face.
I say that, if a sphere can be inscribed within the polyhedron from this center, the two triangles for each edge would be equal.
The plane that is normal to the edge and passes through the center also passes through the sphere-face meeting point $P$, and these two points and the point where the edge and plane meet $N$ make a right triangle with $P$ as the vertex. This gives us a couple of things: no matter what, $P$ is on a circle with $NC$ as a diameter, and $NP$ is the height of the edge's triangle with respect to the edge.
In order for a sphere to be inscribed, both $P$ for any edge must be the same distance from $C$, so the two $P$ are on a circle around $C$. Now we have two circles, and two circles can only intersect in two points, both of which will be the same distance also from $N$ (and indeed from any point along $NC$). So both $NP$ are the same length, and so the height of both triangles are the same, and since they both use the same base, their areas must be equal!
But here, every edge has two colors of face attached, and we know that these two colors add up to different amounts. How do we get unequal things by adding equal values? We cannot! Thus, no matter what center we choose, we are doomed: there must be an edge for which the two associated triangles are of different areas.