Hyper-Kähler structure conjecture for moduli spaces of contact instantons on 3-Sasakian 7-manifolds

Let $(M,g)$ be a $3$-Sasakian $7$-manifold, with Sasakian structures satisfying the quaternionic relations

$$\begin{array}{c} g(\xi_j,\xi_k)=\delta_{jk},\quad [\xi_j,\xi_k] = 2\varepsilon_{jkl}\xi_l,\\ \eta_j(\xi_k) = \delta^{jk}, \quad \Phi^j(\xi^k)=-\sigma_{jkl}\xi^l, \quad \Phi^j\circ\Phi^k-\xi_j\otimes\eta_k =-\sigma_{jkl}\Phi^l-\delta^{jk}\mathbbm{1}. \end{array}$$

Let $\mathcal{M}^\ast$ be the moduli space under consideration. The transverse quaternionic structure conjecture. The transverse quaternionic structure induces a hyper-Kähler structure on $\mathcal{M}^\ast$. This is proposed for the special class of 3-Sasakian 7-manifolds because constructing a hyper-Kähler structure on $\mathcal{M}^\ast$ is difficult in the general 7-dimensional Sasakian case; the source gives no resolution of the claim.