Katz–Langevin–Lee–Sapozhnikov conjecture on the maximum of $V_{F,d}$

Let $F$ be a finite field of characteristic $p$, let $[F:{\mathbb F}_p]$ not be a power of $2$, let $\ell$ be the smallest odd prime divisor of $[F:{\mathbb F}_p]$, and let $d$ be a nondegenerate invertible exponent over $F$. Here $V_{F,d}$ denotes the quantity defined in the surrounding discussion.

Katz–Langevin–Lee–Sapozhnikov conjecture.

$$V_{F,d}\leq \frac{\ell+1}{2\ell}[F:{\mathbb F}_p].$$

The bound is attained in the proposition preceding the conjecture, so the claim is that this observed value is maximal.