The Curtain Model is Not a Quasi-Isometry Invariant of CAT(0) Spaces

Petyt-Spriano-Zalloum recently developed the notion of a \textit{curtain model}, which is a hyperbolic space associated to any CAT(0) space. It plays a similar role for CAT(0) spaces that curve graphs do for mapping class groups of finite-type surfaces. Those authors asked whether this curtain model is a quasi-isometry invariant, namely if quasi-isometric CAT(0) spaces have quasi-isometric curtain models. In this short note, we provide an explicit example answering this question in the negative.