Good $D$ implies good $\varrho$

Let $G\in\mathbb{F}_q^{\ell\times\ell}$ be any kernel, and let $\mathbb{P}\{\mathsf{D}_1=1\}$ denote the probability that the first distance-process value equals $1$. Good $D$ implies good $\varrho$. For any kernel $G\in\mathbb{F}_q^{\ell\times\ell}$,

$$\mathbb{P}\{\mathsf{D}_1=1\}<\ell^{-\varrho}.$$

This bound is intended to control trajectories that do not undergo squaring or higher powering and thereby establish the desired gap-to-capacity behavior. The source presents it as a hope rather than providing a proof, so its resolution is not established here.