The colored interchange process on a graph combines two operations: swapping objects at adjacent vertices and randomly recoloring objects according to some Markov kernel. On specific graphs like complete graphs and cycles, researchers have identified spectral gap formulas that factor nicely, with the spectral gap of the combined process expressible in terms of the spectral gap of the plain interchange process and a separate quantity coming from the coloring mechanism. The open question is whether this factorization structure is a universal phenomenon holding for arbitrary underlying graphs, or whether it breaks down for graphs with more complex geometry such as expanders, trees, or sparse random graphs. A precise version of the problem asks for a general theorem characterizing exactly when and how the spectral gap of the wreath product walk decomposes multiplicatively.