Mozgovoy's degree-independence conjecture for twisted-Higgs Donaldson–Thomas invariants

Let $X$ be a smooth projective curve, let $D$ be a divisor of degree $l$, and let $\cM_D(r,d)$ be the moduli stack of semistable $D$-twisted Higgs bundles of rank $r$ and degree $d$. Set

$$\sH_D(r,d)=(-q^{\frac12})^{-lr^2}[\cM_D(r,d)].$$

For each slope $\tau\in\mathbb R$, define the Donaldson–Thomas invariants $\Om_D(r,d)$ by

$$\sum_{d/r=\tau}\sH_D(r,d)w^rz^d=\operatorname{Exp}\left(\frac{\sum_{d/r=\tau}\Om_D(r,d)w^rz^d}{q-1}\right).$$

Mozgovoy's conjecture. The invariants $\Om_D(r,d)$ are independent of $d$, without requiring $r$ and $d$ to be coprime, and are given by the formula proposed in the cited work. This is a broader conjectural formula for twisted-Higgs Donaldson–Thomas invariants, extending the degree-independence question beyond the coprime case.