Bounded unit-hypercube dimension implies vanishing information rate

Let $\{P_i\}$ be an infinite family of toric codes. For an integral convex polytope $P$, define $M(P)$ to be the largest integer $j$ such that, after a unimodular affine transformation, the unit hypercube $[0,1]^j$ is contained in $P$.

Bounded-hypercube conjecture. If the sequence $\{M(P_i)\}$ is bounded, then

$$R(P_i)\to 0\quad\text{as }i\to\infty.$$

The paper has already shown that unbounded $M(P_i)$ forces the relative minimum distance to tend to zero. Thus this conjecture would rule out good families in the complementary bounded-hypercube case.