Vafa–Witten conjecture for the $L^2$ cohomology of Nakajima's metric

Let $\Hi{n}$ be the Hilbert scheme of $n$ points on the complex plane, equipped with Nakajima's metric, and let $\mathcal{H}^k$ denote the space of $L^2$ harmonic $k$-forms:

$$\mathcal{H}^k=\left\lbrace \alpha\in L^2\left(\Lambda^kT^*\Hi{n}\right),\ d\alpha=d^*\alpha=0 \right\rbrace.$$

Vafa–Witten conjecture.

$$\mathcal{H}^k=\left\lbrace \begin{array}{ll} \{0\}&\ \operatorname{if\ } k\not=2(n-1)=\dim_\mathbb{R}\Hi{n}\\ \operatorname{Im}\left( H_c^k(\Hi{n})\rightarrow H^k(\Hi{n})\right) &\ \operatorname{if\ } k=2(n-1) \end{array} \right..$$

This question, attributed to C. Vafa and E. Witten, concerns the $L^2$ harmonic forms of the moduli space of instantons on noncommutative $\mathbb{R}^4$ and relates analytic $L^2$ cohomology to the image of compactly supported cohomology in ordinary cohomology. The source presents it as a question and gives no evidence of resolution.