The conjectured polynomial-exponential tail bound for Gaussian LU growth with partial pivoting

Let $\widetilde{A}$ be an $n\times n$ matrix with $\|\widetilde{A}\|_2\leq 1$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$. Let $U$ be the upper-triangular matrix obtained from the LU factorization of $A$ with partial pivoting. Partial-pivoting growth conjecture. There exist absolute constants $k_1$, $k_2$, and $\alpha$ such that

$$\Pr\left\{\frac{\|U\|_{\max}}{\|A\|_{\max}}>x+1\right\}\leq n^{k_1}e^{-\alpha x^{k_2}\sigma}.$$

This conjecture addresses the first open problem on smoothed growth factors and is motivated by experiments suggesting substantially smaller tails under partial pivoting than without pivoting.