Kurlberg–Rudnick's limiting distribution conjecture for Hecke matrix elements

Let $A\in SL_2(\mathbb{Z})$, let $U_N(A)$ be its quantization, and let $\\{\psi_j\\}_{j=1}^N$ be a Hecke basis. For $f\in C^\infty(\mathbb{T}^2)$, define

$$F_j^{(N)}=\sqrt{N}\left(\langle\operatorname{Op}_N(f)\psi_j,\psi_j\rangle-\int_{\mathbb{T}^2}f\right).$$

Kurlberg–Rudnick's conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is the distribution of

$$X_f:=\sum_{\nu\ne0}f^\sharp(\nu)\operatorname{tr}(U_\nu),$$

where the $U_\nu$ are independently chosen random matrices in $SU(2)$ with Haar probability measure. This gives a precise probabilistic model for fluctuations of Hecke matrix elements, incorporating the quadratic-form symmetry through $f^\sharp$; its asserted limiting law remains open in the source.