Visual-topology invariance for stabilized $d_L$-metrics

Let $G$ act geometrically on two $\operatorname{CAT}(0)$ spaces $Y$ and $Y'$ whose $d_L$-metrics stabilize. Let $\partial_M^{\kappa}Y$ and $\partial_M^{\kappa}Y'$ denote their $\kappa$-Morse boundaries. Stabilized $d_L$-metric conjecture. The unique $G$-equivariant map

$$\partial_M^{\kappa}Y \rightarrow \partial_M^{\kappa}Y'$$

is a homeomorphism with respect to the visual topology. This would extend the visual-topology invariance theorem from CAT(0) cube complexes with factor systems to the stated class of CAT(0) spaces; the source refers to ongoing work of Petyt, Spriano, and Zalloum for the stabilization condition, and the conjecture remains open.