Independence of refined Higgs invariants from the degree

Let $\mathcal X=(X,M_1,M_2)$ satisfy the conditions of the asymptotic refined ADHM conjecture, and fix $r\geq1$. Let $\overline H(r,e)(y)$ and $\overline H(r,e)(u,v)$ be the alternative refined Higgs sheaf invariants, defined for every $e\in\mathbb Z$.

Degree-independence conjecture. For fixed $r$, both $\overline H(r,e)(y)$ and $\overline H(r,e)(u,v)$ are independent of $e$; equivalently, they have the same value for all pairs $(r,e)$, whether or not $(r,e)$ is coprime.

This conjecture is supported in the paper by direct computations in ranks $r=1,2,3$ and by agreement with known localization and Hodge-polynomial formulas. Its general validity remains open.