Stability of metric Vietoris–Rips persistence for infinite metric spaces

Let $X$ and $Y$ be metric spaces, let $i\geq 0$, and let $d_b$ denote the bottleneck distance between persistent homology modules. Write $\mathrm{PH}_i(\mathrm{VR}^m(X;-))$ and $\mathrm{PH}_i(\mathrm{\check{C}}^m(X;-))$ for the persistent homology modules of the metric Vietoris–Rips and metric Čech filtrations. Infinite-space stability conjecture. The stability inequalities

$$d_b\bigl(\mathrm{PH}_i(\mathrm{VR}^m(X;-)),\mathrm{PH}_i(\mathrm{VR}^m(Y;-))\bigr)\le 2d_{\mathrm{GH}}(X,Y)$$

and

$$d_b\bigl(\mathrm{PH}_i(\mathrm{\check{C}}^m(X;-)),\mathrm{PH}_i(\mathrm{\check{C}}^m(Y;-))\bigr)\le 2d_{\mathrm{GH}}(X,Y)$$

hold even when $X$ and $Y$ are infinite. The finite-space result follows from the identification of metric and simplicial constructions; the infinite-space extension is left conjectural.