Position of aligned shift equivalence between strong shift equivalence and shift equivalence

Ascertain the precise position of aligned shift equivalence of graph C*-correspondences X(A) and X(B) relative to strong shift equivalence and shift equivalence of their adjacency matrices; specifically, determine how aligned shift equivalence compares to these standard relations beyond the facts that it is implied by strong shift equivalence and implies shift equivalence.

Background

Aligned shift equivalence is introduced and studied in the bicategorical framework for C*-correspondences. The authors show that aligned shift equivalence implies shift equivalence and is implied by strong shift equivalence, but its exact relationship to these classical equivalence notions is not fully determined.

Clarifying this position would refine the taxonomy of equivalences for matrices and their associated C*-correspondences, informing classification results for graph C*-algebras and potentially impacting related algebraic frameworks.