Corank bound for intermediate matrices in strong shift equivalences

Prove or disprove that a strong shift equivalence between matrices of corank $k$ can always be realized using only intermediate matrices of corank $k+1$.

Background

In all of the paper’s computational experiments involving two-by-two matrices, the strong shift equivalences found using the specified elementary moves had intermediate matrices of corank one.

This observation motivates a conjectural dimension-growth bound for strong-shift-equivalence paths: starting with matrices of corank kk, one might need no intermediate matrix of corank greater than k+1k+1.

References

Finally, in all our experiments, whenever we found a SSE (using the moves in Section~\ref{proof:bird}) between $2 \times 2$ matrices, this SSE required only intermediate matrices of corank 1. This leads us to conjecture that an SSE between matrices of corank $k$ requires only intermediate matrices of corank $k+1$.

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Conclusion, Section “Conclusion”