Hodge-theoretic generalization of the fibre-restriction generating-function conjecture

Let $\Sigma_{g,\ell}$ be a ruled surface over a Riemann surface $C_g$ of genus $g>0$, with fibre $f$. Let $r\geq 1$, $c_1$ be a first Chern class, and let $H_{r,c_1}(u,v,\tau;f,\Sigma_{g,\ell})$ be the generating function of virtual Hodge functions of the moduli stack of sheaves whose restriction to $f$ is semistable. Then the Hodge-theoretic fibre-restriction conjecture.

$$H_{r,c_1}(u,v,\tau;f,\Sigma_{g,\ell})=\left\{\begin{array}{cl}\displaystyle \frac{i\,(-1)^{r-1}\,\eta(\tau)^{2r(1-g)-3}}{\theta_1(r(u+v),\tau)}\frac{\prod_{j=1}^{r}\theta_1\left(ju+(j-1)v+\frac{1}{2},\tau\right)^g\theta_1\left((j-1)u+jv+\frac{1}{2},\tau\right)^g}{\prod_{k=1}^{r-1}\theta_1(k(u+v),\tau)^2},&\mathrm{if}\,c_1\cdot f=0\mod r,\quad r\geq 1,\\[6pt]0,&\mathrm{if}\,c_1\cdot f\neq 0\mod r,\quad r>1. \end{array}\right.$$

This is a proposed refinement for nonrational ruled surfaces, where the moduli spaces carry off-diagonal Hodge cohomology and virtual Hodge functions retain more information than virtual Poincaré functions. It is suggested by the known Hodge-function formula for vector bundles on $C_g$; the supplied text gives no proof or resolution.