Do–He–Robertson's coefficient-gap conjecture for large-genus Hurwitz numbers

Let $d$ be the degree, let $\mu$ be a ramification profile, and let $C^{\circ}(\mu,m)$ denote the coefficient in the structure theorem

$$H_r^{\circ}(\mu)=\frac{2}{d!^2}\sum_{1\leq m\leq \binom{d}{2}}C^{\circ}(\mu,m)m^{2g-2+d+\ell(\mu)}.$$

Do–He–Robertson's coefficient-gap conjecture. The coefficients satisfy

$$C^{\circ}(\mu,m)=0,\qquad\text{for }\binom{d-1}{2}<m<\binom{d}{2}.$$

This conjectured gap would describe the vanishing of the coefficients immediately below the leading term in the large-genus expansion of connected simple Hurwitz numbers. The source presents it as a conjecture; its resolution is not specified here.