The critical-extremizer characterization of optimal hypercontractive constants on cyclic groups

Let $r_{p,q}(\mathbb{Z}_n)$ denote the optimal hypercontractive constant for $\mathbb{Z}_n$, and let a nontrivial $(p,q)$-critical extremizer mean a nonconstant function attaining the defining inequality at the critical value. For $n\ge 2$ and $1<p<q<\infty$, one may conjecture the following equivalence:

Critical-extremizer characterization.

$$r_{p,q}(\mathbb{Z}_n)=\sqrt{\frac{p-1}{q-1}}$$

if and only if $\mathbb{Z}_n$ does not admit a nontrivial $(p,q)$-critical extremizer.

The conjecture relates equality with the classical two-point hypercontractive bound to the existence of critical extremizers. The surrounding discussion establishes strict inequality and nontrivial critical extremizers for $\mathbb{Z}_3$, but does not resolve the asserted equivalence for general $n$.