Universality of the empirical spectral distribution without sub-Gaussianity

Let $X$ be a random matrix with independent entries conditioned to have general $\delta$-tame margins, and let Theorem (ii) denote the empirical spectral distribution conclusion in the paper.

Non-sub-Gaussian ESD conjecture. The conclusion of Theorem (ii) holds without assuming that the base measure $\mu$ is sub-Gaussian.

The conjecture would extend the empirical spectral distribution theorem beyond sub-Gaussian entry laws. The source proposes deriving it using the logarithmic maximum-entry conjecture and truncation; it does not provide a resolution of this general statement.