Simplify the analysis of higher-order structured factorizations

Determine whether the proofs concerning topological properties, the sufficiency of orthogonal blocks, and related questions for specified numbers of factors, permutation matrices, and block sizes can be substantially simplified.

Background

The paper studies Group-and-Shuffle factorizations with varying numbers of block-diagonal factors, permutations, and block sizes. For two factors with orthogonal blocks, the authors derive a detailed manifold and optimization theory; for three or more factors, they establish selected equivalence and non-equivalence results, construct counterexamples, and analyze when the induced set fails to be a manifold. These arguments become increasingly technical as the factor count, block sizes, and permutations vary.

The unresolved issue is methodological rather than a request to prove a particular classification theorem: the authors explicitly question whether there is a simpler way to establish or refute the relevant topological properties and determine the sufficiency of orthogonal blocks in general higher-order Group-and-Shuffle decompositions.