Polynomial runtime scaling conjecture for reluctant dynamics in the Sherrington–Kirkpatrick model

Let $\mu$ be a “sufficiently nice” probability measure on $\mathbb{R}$ with $\mathbb{E}_{X \sim \mu}X=0$ and $\mathbb{E}_{X \sim \mu}X^2=1$, let $\lambda\in[0,+\infty]$, and suppose that $\sqrt{N}\cdot J$ has i.i.d. entries distributed as $\mu$. Let $T=T(N,\mu,\lambda)$ be the random runtime of reluctant dynamics on an $N\times N$ random matrix $J$. Polynomial runtime scaling conjecture. There exist constants $\alpha=\alpha(\mu,\lambda)>0$ and $\beta=\beta(\mu,\lambda)>0$ such that

$$\frac{T(N,\mu,\lambda)}{\alpha(\mu,\lambda)N^{\beta(\mu,\lambda)}}\to 1$$

in probability as $N\to\infty$. This strengthens prior evidence for polynomial runtime scaling in the classical Sherrington–Kirkpatrick model with Gaussian entries to sufficiently nice entry distributions more generally; the statement remains conjectural.