Sen's monopole $L^2$ cohomology dimension conjecture

Let $M_k^0$ be the hyperkähler moduli space of certain ${\rm SU}(2)$ magnetic monopoles on $\mathbb R^3$, and let $\widetilde{M}_k^0$ be its universal cover. Write $\Ha^d(\widetilde{M}_k^0)$ for the space of $L^2$ harmonic $d$-forms, let $\phi(k)=\sum_{i=1}^{k}\delta_{1 (i,k)}$ be the Euler $\phi$ function, and set $mid=2k-2$, half the dimension of $\widetilde{M}_k^0$. Sen's conjecture. The dimension is

$$\dim\left(\Ha^d(\widetilde{M}_k^0)\right)=\left\{\begin{array}{ll} 0 & d\neq mid \\ \phi(k) & d=mid \end{array} \right..$$

This prediction arises from $S$-duality in four-dimensional $N=4$ supersymmetric Yang--Mills theory, where the $L^2$ harmonic forms are interpreted as bound states. The supplied text gives no mathematical resolution, so the conjecture is recorded as open.