Generalized collapsing-map conjecture for function-bounded quasi-isometries

Suppose a generalized quasi-isometry has bounds given by a function, and suppose there are a generalized $(b,c)$-metric and a generalized collapsing map whose bounds are given by the same function.

Generalized collapsing-map conjecture. Then the generalized form of Theorem 2 and Corollary 2.1 holds.

The conjecture proposes that the preservation results for collapsing maps extend from linear bounds to bounds controlled by a common function. The source gives no resolution or further hypotheses.