Conjecture on the smallest bulk gap for real Wigner matrices

Fix $\alpha>0$. Let $\mu_1\leq\cdots\leq\mu_N$ be the eigenvalues of a GOE matrix, and let $\lambda_1\leq\cdots\leq\lambda_N$ be the eigenvalues of a real generalized Wigner matrix $(H_N)$ with uniform $p$-support, where $p\geq2$. Smallest-gap convergence conjecture. For some sufficiently large $p$, for all sufficiently large $N$,

$$d_{\mathrm K}\biggl(\min_{\alpha N\leq k\leq(1-\alpha)N}N^{3/2}(\lambda_{k+1}-\lambda_k),\min_{\alpha N\leq k\leq(1-\alpha)N}N^{3/2}(\mu_{k+1}-\mu_k)\biggr)=o(1).$$

A stronger conjecture states that $p=2$ is enough. In particular, together with the cited GOE smallest-gap law, the conjecture predicts a constant $C=C(\alpha)$ such that, for every $x\geq0$, the distribution function converges to $\int_0^x2u e^{-u^2}\,du$. The source gives no resolution status.