Vafa–Witten S-duality conjecture for SU(r) and PSU(r) partition functions

Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r$ be prime and let $c_1\in H^2(X,\mathbb{Z})$ be algebraic. Let $q=\exp(2\pi\sqrt{-1}\\,\tau)$, where $\tau$ lies in the upper half plane, and regard the two partition functions as Fourier expansions of meromorphic functions of $\tau$. Vafa–Witten S-duality conjecture. The functions satisfy

$$\mathsf{VW}^{\mathrm{SU}(r)}_{X,c_1}(-1/\tau)=(-1)^{(r-1)\chi(\mathcal{O}_X)}\left(\frac{r\tau}{\sqrt{-1}}\right)^{-e(X)/2}\mathsf{VW}^{\mathrm{PSU}(r)}_{X,c_1}(\tau).$$

The conjecture was checked in the source for $r=2,3,5$ using the corresponding conjectural closed expressions for the universal functions; it remains open in general.