Linear-size conjecture for the distilled Vietoris–Rips filtration of manifold samples

Let $X$ be a finite sample of points from a $k$-dimensional manifold $M$ embedded in Euclidean space, and let $n=|X|$. The filtration $\mathcal{D}_{\infty}^{1}(X)$ is the distilled Vietoris–Rips filtration in degree one.

Linear-size conjecture. The filtration $\mathcal{D}_{\infty}^{1}(X)$ consists of $O(kn)$ simplices.

This conjecture predicts linear memory usage in the sample size, up to the manifold dimension, for the distilled filtration on point clouds sampled from manifolds. The supplied text does not state whether the claim has been proved or disproved.