Counting diverging-root eigenstates near special temperatures

Fix the magnetic field in the XXZ chain and let $T_m=\frac{h}{2m\gamma}$ with $m\in\mathbb{N}$. Consider eigenstates in the sector with spin $s$, equivalently with $M=\frac{N}{2}-s$ Bethe roots, at $T\sim T_m$. Diverging-root counting conjecture. If $s+1\le m\le\frac{N}{2}$, there are $\binom{N}{N/2-m}$ eigenstates, out of the total $\binom{N}{M}$ states in the sector, with $m-s$ diverging roots. This characterization is based on numerical investigations near the temperatures where roots move to infinity; the counting assertion is not established in the supplied text.