Conjecture on the most stably singular initial condition for Dyson Brownian motion

Let $\lambda_1,\ldots,\lambda_n$ be the initial eigenvalues for Dyson Brownian motion, with parameter $\beta\geq 0$, and let $R^{(1)}_\beta(0)$ denote the one-point function at time $t=1$ evaluated at $0$. The initial eigenvalues are allowed to vary over all initial conditions.

Most stably singular initial-condition conjecture. Among all initial conditions for Dyson Brownian motion, those maximizing $R^{(1)}_\beta(0)$ are precisely

$$\lambda_1=\cdots=\lambda_n=\begin{cases}0&\text{if }\beta\leq 1\text{ or }n\text{ is odd},\\ \pm x_{n,\beta}\neq 0&\text{if }\beta>1\text{ and }n\text{ is even}. \end{cases}$$

Here $x_{n,\beta}$ is the nonzero value appearing in the even-$n$, $\beta>1$ case.

The conjecture identifies the initial configuration that maximizes the density of eigenvalues at the origin after unit-time Dyson Brownian evolution. The supplied excerpt presents it as a conjecture based on the results in the paper and further fragmentary evidence; no resolution is given there.