Factorization conjecture for quasi-isometries through collapsing maps

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let a quasi-isometry mean a map satisfying the usual multiplicative and additive coarse bounds. A collapsing map is a map obtained by collapsing a subset via an equivalence relation, and a Lipschitz map is a map with a linear distance bound.

Factorization conjecture. Any quasi-isometry can be expressed as the composition of a collapsing map and Lipschitz map.

This conjecture proposes that quasi-isometries, including canonical examples such as the floor function, admit a factorization into elementary collapsing and Lipschitz operations. The source gives no resolution or further hypotheses.