Monotone edge-shrinking and vertex-expansion conjecture for polytopes

Let $P,Q\subset\mathbb{R}^d$ be polytopes with isomorphic edge-graphs. Assume that $Q$ contains the origin in its interior, edges in $Q$ are not longer than the corresponding edges in $P$, and vertex-origin distances in $Q$ are not smaller than the corresponding distances in $P$. Monotone edge-shrinking and vertex-expansion conjecture. Then $P\cong Q$. The conjecture removes the requirement that the polytopes be combinatorially equivalent, retaining only isomorphism of their edge-graphs; the source gives no resolution.