The optimal scalar-fluctuation upper-bound conjecture for random band matrices

Let $A$ be the random band matrix model in the paper, with $D_k$ and $B_k$ defined by Gaussian elimination, let $S_k=\lVert D_k\rVert$, and let $\bar D_k=\lVert D_k\rVert^{-1}D_k$. Write $\mathbf{V}(X\mid\mathcal{G})$ for conditional variance and $\mathbf{E}$ for expectation.

Scalar-fluctuation upper-bound conjecture. If $\varepsilon>0$ is small, $|\lambda|<\varepsilon$, $M\geq C_\varepsilon$, and $1<k<N$, then

$$\mathbf{E}\bigl(\mathbf{V}(\log S_k\mid D_{k-1},\bar D_k,D_{k+1},B_{k-1},B_k)\bigr)\leq M^{-3+\varepsilon}.$$

This conjecture would show that the scalar fluctuations used in the paper are of order at most $M^{-3+\varepsilon}$ near zero energy, supporting the claim that the $1/4$ localization exponent is optimal for arguments based on these fluctuations. The source gives no resolution status.