Vafa–Witten's $L^2$ harmonic form dimension conjecture for instanton moduli spaces

Let $M^{k,c_1}_\phi$ be the smooth completion of the moduli space of ${\rm U}(n)$ Yang--Mills instantons of first Chern class $c_1$, energy $k$, and framing $\phi$ on a four-dimensional Kronheimer ALE space. Write $\Ha^d(M^{k,c_1}_\phi)$ for its space of $L^2$ harmonic $d$-forms, and let $mid$ be half the dimension of $M^{k,c_1}_\phi$. Vafa--Witten's conjecture.

$$\dim\left(\Ha^{d}(M^{k,c_1}_\phi)\right)=\left\{ \begin{array}{ll} 0 & d\neq mid\\ \dim\left({\rm im}(H_{cpt}^{mid}(M^{k,c_1}_\phi)\rightarrow H^{mid}(M^{k,c_1}_\phi))\right) & d=mid\end{array} \right..$$

The conjecture is another prediction of $S$-duality, now for instanton moduli spaces on ALE manifolds. The supplied text gives no resolution, so its status is open.