Kurlberg–Rudnick's IID conjecture for quadratic-form-indexed Hecke matrix elements

Let $A\in SL_2(\mathbb{Z})$ satisfy $A\equiv I\pmod 2$, let $N$ tend to infinity through primes, and let $\\{\psi_j\\}_{j=1}^N$ be a Hecke basis. Define $Q(n)=\omega(n,nA)$ and, for $\nu\in\mathbb{Z}$ for which $Q(n)=\nu$, define

$$Y_\nu(\psi)=\sqrt N(-1)^{n_1n_2}\langle T_N(n)\psi,\psi\rangle.$$

Quadratic-form IID conjecture. For every $\nu\in\mathbb{Z}$, $Y_\nu(\psi)$ has limiting distribution equal to that of $\operatorname{tr}(U_\nu)$, where $U_\nu$ is Haar-random in $SU(2)$; moreover, the sequence

$$\dots,Y_{-3}(\psi),Y_{-2}(\psi),Y_{-1}(\psi),Y_1(\psi),Y_2(\psi),Y_3(\psi),\dots$$

converges to a sequence of independent identically distributed random variables. This is a symmetry-reduced formulation of the Hecke fluctuation conjecture, with the quadratic form indexing the natural symmetry classes; the limiting IID assertion remains open.