Optimality conjecture for Levenshtein-bound refinements

Let $q\geq 3$ be fixed. Let $s_q$ be a constant, let $s\in[-1,s_q)\cap T_n$, and interpret $s\in T_n$ in the setting of the paper's code bounds, with $d/n=(1-s)/2$. Theorem 1 provides the relevant upper bound and the paper's refinements are the bounds being compared.

Optimality conjecture. There exists a constant $s_q$ such that whenever $s\in[-1,s_q)\cap T_n$, the refinements described above are the best bounds that can be obtained from Theorem 1.

The conjecture asserts asymptotic-range optimality of the proposed refinements for sufficiently large relative distance. The supplied passage gives computational and comparative evidence but no resolution, so the conjecture remains open.