The cube conjecture for pattern-avoiding polytopes $P_n(132,213)$

Let $P_n(132,213)$ be the pattern-avoiding polytope associated with the patterns $132$ and $213$. A cube conjecture asserts that, for every $n$, $P_n(132,213)$ is a combinatorial cube and has normalized volume

$$2^{n-1}n^{n-3}.$$

The conjecture is presented as analogous to Proposition~, but the authors state that they have been unable to prove it.