The longest Vietoris-Rips persistence interval conjecture for metric manifolds

Let $M$ be a closed connected $n$-dimensional metric manifold. Let $\mathrm{barc}^{\mathrm{VR}}_k(M;\mathbb{F})$ denote the Vietoris-Rips persistence barcode in degree $k$, and let $I_n^M$ be the distinguished interval in degree $n$ associated with $M$.

Longest-interval conjecture. For every $I\in\mathrm{barc}^{\mathrm{VR}}_k(M;\mathbb{F})$ and every $k\geq 1$,

$$\mathrm{length}(I)\leq\mathrm{length}(I_n^M).$$

This conjecture proposes that the distinguished top-dimensional interval controls the lengths of all positive-degree Vietoris-Rips persistence intervals. The source motivates it from the circle case but gives no resolution in general.