Singularity conjecture for random matrices with anti-concentrated entries

Let $\mathcal{D}$ be a distribution over $\mathbb{R}$ and set $p=\|\mathcal{D}\|_{\infty}$, where $\|\mathcal{D}\|_{\infty}=\max_x \mathcal{D}(x)$. Let $M$ be a random $n \times n$ matrix with independent entries from $\mathcal{D}$. Singularity conjecture. There is an absolute constant $c>0$ such that

$$\Pr[M \text{ is singular}] \le p^{cn}.$$

This conjecture proposes that the singularity probability is controlled by the anti-concentration of the entry distribution, extending the paper's result for entries uniformly distributed on $\{-m,\ldots,m\}$. Its validity for general distributions over $\mathbb{R}$ remains open.