Bounded Minkowski length implies vanishing information rate

Let $\{P_i\}$ be an infinite family of toric codes. For each integral convex polytope $P$, let $L(P)$ denote its Minkowski length, and let $R(P)$ denote the information rate of the associated toric code.

Bounded-Minkowski-length conjecture. If the sequence $\{L(P_i)\}$ is bounded, then

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

This is the reformulation, in terms of Minkowski length, of the earlier conjecture involving bounded unit-hypercube dimension. The supplied text provides no resolution.