Conjecture on the optimal speed of bulk gap convergence for Wigner matrices

Let $(H_N)$ be a generalized Wigner ensemble with uniform $p$-support, and let $\mu_1\leq\cdots\leq\mu_N$ be the eigenvalues of a GOE matrix while $\lambda_1\leq\cdots\leq\lambda_N$ are the eigenvalues of $H_N$. For fixed $\alpha,\varepsilon>0$, write $d_{\mathrm K}$ for the Kolmogorov distance between probability laws. Optimal speed of convergence conjecture. There \exists $p=p(\varepsilon)$ sufficiently large such that, for all sufficiently large $N$ and uniformly in $\alpha N\leq k\leq(1-\alpha)N$,

$$d_{\mathrm K}\bigl(N(\lambda_{k+1}-\lambda_k),N(\mu_{k+1}-\mu_k)\bigr)\leq N^{-1+\varepsilon}.$$

Moreover, a stronger conjecture states that $p=2$ is enough. The claim would improve the currently stated $N^{-1/2+\varepsilon}$ convergence rate to the conjecturally optimal $N^{-1+\varepsilon}$ scale; the source gives no resolution status.