Quadratic optimal-time conjecture for the quantum Ornstein–Uhlenbeck semigroup

For $2\leq p<8$, let $t'_p$ denote the optimal time and let $\beta$ be the parameter of the quantum Ornstein–Uhlenbeck semigroup. There are positive constants $\widetilde{c}_2(\beta)$ and $\widetilde{C}_2(\beta)$ depending only on $\beta$. Quadratic optimal-time conjecture. The optimal time satisfies

$$\widetilde{c}_2(\beta)(p-1)^2\leq e^{2\tau t'_p}\leq \widetilde{C}_2(\beta)(p-1)^2.$$

This conjecture predicts the sharp quadratic dependence of the optimal hypercontractive time on $p-1$ in the stated range. The preceding corollary establishes an upper bound of the same order, while the corresponding lower bound remains conjectural.