Invariance of the SU(r) partition function under tensoring

Let $(S,H)$ be a smooth polarized surface, and let $c_1,b3\in H^2(S,\mathbb Z)$ be algebraic classes. The $a5\mathrm{SU}(r)$ Vafa13Witten partition function is denoted by $\mathsf{Z}^{\mathrm{SU}(r)}_{c_1}(q)$.

Tensoring invariance conjecture. For any such $(S,H)$, $c_1$, and $b3$, one has

$$\mathsf{Z}^{\mathrm{SU}(r)}_{c_1}(q)=\mathsf{Z}^{\mathrm{SU}(r)}_{c_1+r\gamma}(q).$$

This invariance is immediate when Gieseker and $bc$-stability coincide, since tensoring by $\mathcal O_S(\gamma)$ induces an isomorphism of moduli spaces; its general validity is asserted here.