The logarithmic maximum-entry conjecture for tame random matrices

Let $(\mathbf{r},\mathbf{c})$ be an $m\times n$ $\delta$-tame margin for a base measure $\mu$, and let $X\sim\lambda_{\mathbf{r},\mathbf{c}}$ be the corresponding margin-conditioned random matrix.

Logarithmic maximum-entry conjecture. With high probability,

$$\max_{ij}X_{ij}\leq C\log(m+n)$$

for some constant $C=C(\mu,\delta)>0$.

This conjecture extends the known logarithmic maximum-entry behavior from constant linear margins and classical Lebesgue or counting measures to general base measures and tame margins. The source notes that it remains unresolved in this broader setting.