The metric thickening finite-support homotopy equivalence conjecture

Let $X$ be a metric space and let $K$ be a pseudo-simplicial complex with vertex set $V(K)=X$. Define the metric thickening

$$\mathcal{K}=\left\{\mu\in\mathcal{P}(X)\mid \operatorname{supp}(\mu)\in S(K)\right\}$$

and the finite metric thickening

$$\operatorname{finK}^{\mathrm{m}}=\left\{\mu\in\mathcal{P}(X)\mid \operatorname{supp}(\mu)\in S(K),\ \#\operatorname{supp}(\mu)<\infty\right\}.$$

Metric thickening finite-support homotopy equivalence conjecture. The inclusion induces a homotopy equivalence

$$\mathcal{K}\simeq \operatorname{finK}^{\mathrm{m}}.$$

This asserts that allowing probability measures with arbitrary supports does not change the homotopy type of the metric thickening. The supplied text gives no evidence that the statement has been proved or disproved, so its status remains open.