Doubly refined wallcrossing and multicover conjecture

Let $u$ and $v$ be formal variables, with formal square roots $u^{1/2}$ and $v^{1/2}$, and define

$$[n]_{(u,v)}=\frac{(uv)^{n/2}-(uv)^{-n/2}}{(uv)^{1/2}-(uv)^{-1/2}}.$$

Let $A_\delta(r,e)(u,v)$ and $H(r,e)(u,v)$ denote doubly refined residual ADHM and Higgs sheaf invariants, and let $A_\delta(r,e)(y)$ and $H(r,e)(y)$ be their singly refined specializations.

Doubly refined wallcrossing and multicover conjecture. Under the same conditions as the preceding refined wallcrossing conjecture, there exist doubly refined invariants in $\mathbb Q(u^{1/2},v^{1/2})$ such that

$$A_\delta(r,e)(u,u)=A_\delta(r,e)(u),\qquad H(r,e)(u,u)=H(r,e)(u),$$

with Laurent-polynomial integrality in the noncritical and coprime cases. The wallcrossing formulas are obtained from the singly refined formulas by replacing $A_\delta(\gamma_i)(y)$, $H(\gamma_i)(y)$, and $[e_i-r_i(g-1)]_y$ by their doubly refined versions. There are also alternative invariants $\overline H(r,e)(u,v)$ satisfying the corresponding multicover formula obtained by the same substitutions.