The projective-space metric-thickening stability conjecture

Let $\mathbb{R}\mathrm{P}^n$ be real projective $n$-space, and let $\mathrm{VR}^m$, $\mathrm{VR}$, and $\mathrm{VR}^m_\leq$ denote the metric, Vietoris–Rips, and measure Vietoris–Rips thickenings used in the paper. At scale $1/6$, $\mathrm{VR}^m_\leq(\mathbb{R}\mathrm{P}^n;1/6)$ has the homotopy type of a $(2n+1)$-dimensional CW complex.

Projective-space stability conjecture. There exists a sufficiently small $\varepsilon>0$ such that, for every $0<\delta<\varepsilon$, both $\mathrm{VR}^m(\mathbb{R}\mathrm{P}^n;1/6+\delta)$ and $\mathrm{VR}(\mathbb{R}\mathrm{P}^n;1/6+\delta)$ have the homotopy type of $\mathrm{VR}^m_\leq(\mathbb{R}\mathrm{P}^n;1/6)$.

If true, this would identify the homotopy type immediately above the critical scale $1/6$ and relate the metric and ordinary Vietoris–Rips constructions to the measure thickening.