Closure equivalence conjecture for metric thickenings

Let $M=(X,K,\phi)$ be a simplicial metric thickening, with metric thickening $\mathcal{M}\subseteq\mathcal{P}(X)$. Let $\hat{M}$ be the closure of $\mathcal{M}$ in $\mathcal{P}(X)$, and let $\bar{M}$ be its closure in the finitely-supported metric thickening $\mathcal{FX}$. In the notation used for the associated simplicial complex, these spaces are denoted $\hat{K}$ and $\bar{K}$, respectively. Closure equivalence conjecture. The metric thickening and its closure in the finitely-supported space are homotopy equivalent:

$$\mathcal{K}\simeq\bar{K}.$$

The surrounding text presents this as an unresolved conjecture intended to connect the infinitely-supported space with the theorem that $\bar{M}$ and $\hat{M}$ are homotopy equivalent. It is meant to show that the infinitely-supported space has the same homotopy type independently of the closure properties of the original space.