The GOE nodal-count central limit conjecture

Let $A$ be a random $n\times n$ matrix drawn from the Gaussian Orthogonal Ensemble. For fixed $k$, define the normalized nodal count by

$$\overline{\phi(A,k)}:=\frac{\phi(A,k)-\mathbb{E}[\phi(A,k)]}{\sqrt{\mathrm{Var}(\phi(A,k))}}.$$

GOE nodal-count central limit conjecture. For every $k\in[n]$,

$$\mathrm{dist}\left(\overline{\phi(A,k)},N(0,1)\right)=o(1),$$

as $n$ grows, uniformly in $k$, for any reasonable distance between random variables. The conjecture concerns Gaussian fluctuations of GOE nodal counts; the paper reports numerical evidence but does not provide a proof in the supplied text.