Kurlberg–Rudnick rate conjecture for Hecke quantum unique ergodicity

Let $p$ be prime, let $\rho_{1/p}(A)$ be the quantized operator, and let ${\rm T}_A$ be its Hecke torus. For a character $\chi:{\rm T}_A\@ifnextchar^{\t@@}{\t@@^{}}\mathbb C^*$, let ${\cal H}_\chi$ be the corresponding Hecke eigenspace, choose a unit eigenstate $\Psi\in{\cal H}_\chi$, and define its Wigner distribution by

$$\mathcal{W}_\chi(f)=\langle\Psi\mid\pi_{1/p}(f)\Psi\rangle.$$

Here $f\in C^\infty(\mathbb T)$, $|\omega|$ is the invariant measure on $\mathbb T$, and $C_f$ depends only on $f$. Rate conjecture. The following bound holds:

$$\left|\mathcal{W}_\chi(f)-\int_{\mathbb T}f|\omega|\right|\leq\frac{C_f}{p^{1/2}}.$$

This strengthens the previously established $p^{-1/4}$ estimate and is a conjectural optimal-rate form of Hecke quantum unique ergodicity.