Phase transition in optimal hypercontractivity

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$t_{\mathrm{o}\mathrm{p}\mathrm{t}}(2,q)=\frac{1}{2}\log (q-1)\text{ if and only if }q\ge q_0,$$ whereas the strict inequality $t_{\mathrm{o}\mathrm{p}\mathrm{t}}(2,q)>\frac{1}{2}\log (q-1)$ holds for $2<q<q_0$.