Conjectural generating function for fibre-semistable sheaves on Hirzebruch surfaces

Let $\Sigma_\ell$ be a Hirzebruch surface with generic fibre $f$, let $r\geq 1$, and let $c_1\in H_2(\Sigma_\ell,\mathbb{Z})$. Write $H_{r,c_1}(z,\tau;f)$ for the generating function of the virtual Poincaré functions of moduli stacks of sheaves whose restriction to $f$ is semistable. Then the fibre-restriction generating-function conjecture.

$$H_{r,c_1}(z,\tau;f)=\left\{\begin{array}{cl}\displaystyle \frac{i\,(-1)^{r-1}\,\eta(\tau)^{2r-3}}{\theta_1(2z,\tau)^2\,\theta_1(4z,\tau)^2\dots\theta_1((2r-2)z,\tau)^2\,\theta_1(2rz,\tau)},&\mathrm{if}\,c_1\cdot f=0\mod r,\quad r\geq 1,\\[6pt]0,&\mathrm{if}\,c_1\cdot f\neq 0\mod r,\quad r>1. \end{array}\right.$$

The formula extends the generating function for virtual Poincaré functions of vector-bundle stacks on the fibre and is motivated by known results for ranks one and two. A derivation from the relevant moduli stacks is not provided for ranks $r\geq 3$, although consistency checks are given for ranks $3$ and $4$ and for other Hirzebruch surfaces.