Loop-space conjecture for the space of core metrics on spheres

For a smooth manifold $\operatorname{M}^n$, fix an embedded disk $\operatorname{D}^n\subseteq \operatorname{M}^n$. Let $\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{M}^n)$ denote the space of core metrics on $\operatorname{M}^n$. In particular, combining core metrics on spheres gives a binary operation

$$\mu:\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n)\times\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n)\longrightarrow\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n).$$

Loop-space conjecture. The space $\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n)$ equipped with $\mu$ is an associative $H$-space and is homotopy equivalent to a loop space. This conjecture proposes that the product structure on core metrics has the same loop-space behavior as analogous product structures on spaces of positive scalar curvature metrics and metrics of positive Ricci curvature.