Refined multicover conjecture for Higgs sheaf invariants

Let $(r,e)\in\mathbb Z_{\geq1}\times\mathbb Z$, and let $H(r,e)(y)$ and $\overline H(r,e)(y)$ be refined Higgs sheaf invariants related by the preceding motivic-invariant conjecture. For $n\in\mathbb Z$, set $[n]_y=(y^n-y^{-n})/(y-y^{-1})$.

Refined multicover conjecture. One has

$$H(r,e)(y)=\sum_{\substack{k\geq1\\ k\mid r,\ k\mid e}}\frac{1}{k[k]_y}\,\overline H\!\left(\frac r k,\frac e k\right)(y^k).$$

This is the refined relation between the Higgs invariants and their alternative Kontsevich–Soibelman-type invariants; the source explains that it specializes to equality for primitive invariants and is motivated by refined wallcrossing.