The modulus-of-continuity conjecture for topologies on Urysohn isometry groups

Let $\mathcal{R}$ be a distance monoid, let $\mathcal{U}$ be an $\mathcal{R}$-Urysohn space, and write $G=\operatorname{Isom}(\mathcal{U})$. Let $\tau_{st}$ denote the stabilizer topology, and let an $\mathcal{R}$-modulus of continuity mean a function of the type defined in the paper that determines a topology $\tau_f$ on $G$.

Modulus-of-continuity conjecture. Every group topology on $G$ strictly coarser than $\tau_{st}$ is of the form $\tau_f$ for some $\mathcal{R}$-modulus of continuity.

This conjecture proposes a parametrization of all strictly coarser group topologies by moduli of continuity, motivated by their interpretation through generalized bi-Katetov maps and types of pairs of copies of the Urysohn space. Its general validity is open.