Quasi-Möbius homeomorphisms of Morse boundaries are induced by quasi-isometries

Let $X$ and $Y$ be proper, cocompact CAT(0) spaces, and let $f:\partial_*X\to\partial_*Y$ be a homeomorphism, where $\partial_*X$ and $\partial_*Y$ denote their Morse boundaries. A homeomorphism is 2-stable if it preserves the relevant 2-stable pairs, and quasi-Möbius if it distorts cross-ratios in the prescribed quasi-Möbius sense. Quasi-Möbius homeomorphism conjecture. Every quasi-Möbius homeomorphism

$$f:\partial_*X\to\partial_*Y$$

is 2-stable and hence is induced by a quasi-isometry $h:X\to Y$. The preceding theorem establishes the converse implication under the stated hypotheses, while this conjecture would show that quasi-Möbius behavior alone also forces 2-stability and quasi-isometric origin.