Real-part sampling conjecture for Benjamin–Ono bulk eigenvalue branches

Let $S_U$ and $S_L$ denote the upper and lower branches of the bulk eigenvalues, with real parts indexed increasingly by $\mu_{U,k}(t)$ and $\mu_{L,k}(t)$, and let $t_b$ be the breaking time. Real-part sampling conjecture. For all $t\ge0$ and sufficiently small $\epsilon$, these real parts are approximate samplings of $\epsilon$-independent functions $\mu_U:(0,1)\times[0,\infty)\to\mathbb R$ and $\mu_L:(0,1)\times(t_b,\infty)\to(X^-(t),X^+(t))$, with

$$\mu_{U,k}(t)=\mu_U(y_k,t)+\mathcal O(\epsilon^2\mu_U'(y_k,t)),\qquad y_k=\frac{k-\frac12}{N_U(\epsilon)},$$

and, for $t>t_b$,

$$\mu_{L,k}(t)=\mu_L(y_k,t)+\mathcal O(\epsilon^2\mu_L'(y_k,t)),\qquad y_k=\frac{k-\frac12}{N_L(\epsilon)}.$$

The functions have the invertibility, smoothness, edge-growth, and uniformity properties stated in the conjecture, including $\epsilon N_U(\epsilon)$ and $\epsilon N_L(\epsilon)$ having finite nonzero limits in the indicated regimes. This is a detailed numerical conjecture about the limiting distribution of bulk eigenvalue real parts; no proof or resolution is supplied.