Lower-branch imaginary-part sampling conjecture for Benjamin–Ono eigenvalues

Let $t>t_b$ and let $\nu_{L,k}(t)$ be the imaginary parts of the lower-branch eigenvalues. Lower-branch imaginary-part conjecture. There is an $\epsilon$-independent function $\nu_L:(0,1)\times(t_b,\infty)\to(0,\infty)$ such that

$$\nu_{L,k}(t)=\epsilon\nu_L(y_k,t)(1+o(1)),\qquad y_k=\frac{k-\frac12}{N_L(\epsilon)},$$

uniformly in $k$. The scale $\delta(\epsilon)$ from the upper-branch conjecture additionally satisfies

$$\int_0^\epsilon\frac{\delta(y)}{y}\,\mathrm dy<\infty.$$

The function $\nu_L$ has the stated positive lower bound, left-edge behavior, and $C^1$ regularity, uniformly for bounded $t>t_b$ away from $t_b$. This numerical conjecture remains open.