Götsche–Kool numerical conjecture for rank 2 sheaf invariants

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $\underline\beta\in\mathbb Z^4$ satisfy $\beta_1\equiv\beta_2\pmod 2$ and $\beta_3\ge\beta_4-3$, and let $(\gamma_1,\gamma_2)\in\mathbb Z^2$. Götsche–Kool's numerical conjecture. For every $n<\frac12(\beta_1-\beta_2)+2\beta_4$, the coefficient of $s^0p^{4n-\beta_1-3\beta_4}$ in the displayed sum of the two $\mathsf A^a$-series equals the coefficient of $p^{4n-\beta_1-3\beta_4}$ in $\psi^a_{\gamma_1,\gamma_2,\beta_3,\beta_4}(p,u^a)$, with both expressions exactly as given in the source statement. This numerical prediction generalizes the cited conjecture of Götsche–Kool and is tested through coefficient calculations in the paper, but is not proved in general.