Simplicial collapse for chromatic alpha filtrations under refinement

Determine whether, for any finite point cloud X in general position and any colourings ν and μ of X with ν a refinement of μ, there exists, for every filtration parameter r ≥ 0, a simplicial collapse from the chromatic alpha filtration Alpha_r(X,ν) to Alpha_r(X,μ).

Background

The paper proves that for refinements of colourings ν ≼ μ, the chromatic alpha filtrations Alpha_r(X,ν) and Alpha_r(X,μ) have the same simple homotopy type for all r ≥ 0. This is achieved via collapses in related constructions (chromatic Delaunay–Čech) and by Morse-theoretic arguments, but it does not directly yield collapses between the chromatic alpha filtrations themselves.

The authors provide a counterexample showing that a single generalized discrete Morse function cannot induce such collapses simultaneously for all r, leaving open whether a collapse exists individually at each filtration level r. Establishing per-level collapses would strengthen the known equivalence from simple homotopy type to explicit combinatorial reductions of the complexes at each scale.