Myrberg-set invariance for geometric CAT(0) actions

Let $G$ act geometrically on two proper CAT(0) spaces $X$ and $Y$, and let the corresponding Myrberg sets be the subsets of their respective boundaries consisting of Myrberg points. Myrberg-set invariance conjecture. There exists a $G$-equivariant homeomorphism between the corresponding Myrberg sets. This would show that the Myrberg set is independent, up to $G$-equivariant homeomorphism, of the proper CAT(0) space on which $G$ acts geometrically; the supplied text presents it as a question supported by positive evidence, with no resolution stated.