Vanishing-range conjecture for Hurwitz-number coefficients with at least two fixed points

Let $d\geq10$ and let $\nu$ be a partition of $d$ with $m_1(\nu)\geq2$. Let $z_\nu$ be the standard centralizer factor, and let $b^X_\nu(\mu^{(1)},\dots,\mu^{(s)},m)$ denote the coefficients from Theorem~. Vanishing-range conjecture for $m_1(\nu)\geq2$. These coefficients vanish when

$$\frac{d!m_1(\nu)}{d\cdot z_\nu}<m<\frac{d!}{z_\nu}$$

and when

$$\frac{d!(m_1(\nu)-1)}{(d-1)z_\nu}<m<\frac{d!m_1(\nu)}{d\cdot z_\nu},$$

and

$$b^X_\nu\left(\mu^{(1)},\dots,\mu^{(s)},\frac{d!m_1(\nu)}{d\cdot z_\nu}\right)=-d^{2-2g(X)-s}\prod_{i=1}^s m_1(\mu^{(i)}).$$

Moreover, if $\nu\neq(2^{d/2-1},1^2)$, then

$$b^X_\nu\left(\mu^{(1)},\dots,\mu^{(s)},\frac{d!(m_1(\nu)-1)}{(d-1)z_\nu}\right)=(d-1)^{2-2g(X)-s}\prod_{i=1}^s(m_1(\mu^{(i)})-1).$$

The conjecture is presented as a coefficient-level extension of the character-ratio bounds and is part of the paper's proposed large-genus asymptotic picture; no general proof is supplied.