Strong enumeration conjecture for q-ary codes

Fix an integer $q\ge 2$. A $q$-ary code of length $n$ and distance $d$ is a subset of $[q]^n$ whose elements are pairwise separated by Hamming distance at least $d$, and let $A_q(n,d)$ denote the maximum size of such a code. Strong enumeration conjecture for q-ary codes. For every fixed $c>0$, the number of $q$-ary codes of length $n$ and distance $d$ is

$$2^{O(A_q(n,d))}$$

whenever

$$d<(1-q^{-1}-c)n.$$

The conjecture would give a substantially stronger enumeration bound than the paper's proved results. Its main obstacle is the apparent lack of strong supersaturation estimates showing that sets much larger than $A_q(n,d)$ contain many pairs at distance at most $d$; the source presents it as likely quite difficult and gives no resolution.