Dolorfino et al.'s bounded-hypercube conjecture for toric-code families

Let $q$ be a fixed prime power, and let $\{P_i\}_i$ be an infinite family of toric codes: a sequence of nonempty integral convex polytopes with $P_i\subseteq[0,q-2]^{n_i}\subseteq\mathbb{R}^{n_i}$, $n_i\to\infty$, and convergent sequences $\{d(P_i)\}_i$ and $\{R(P_i)\}_i$. For an integral convex polytope $P$, define

$$M(P):=\max\{m\mid\exists\text{ a unimodular affine transformation }A\text{ such that }A([0,1]^m)\subseteq P\}.$$

Dolorfino et al.'s bounded-hypercube conjecture. If $\{M(P_i)\}_i$ is bounded, then

$$R(P_i)\to0\quad\text{as }i\to\infty.$$

This conjecture would rule out good infinite families in the remaining case not covered by the result that unbounded $M(P_i)$ forces $d(P_i)\to0$; together with that result, it would imply that no good infinite family of toric codes exists.