Similarity versus strong shift equivalence for primitive two-by-two matrices

Determine whether any two $2\times2$ primitive nonnegative integer matrices that are similar over $\mathbb{Z}$ must be strong shift equivalent.

Background

Similarity over the integers is a necessary-equivalence indicator closely related to shift equivalence, and several sufficient criteria are known for primitive two-by-two matrices under determinant and trace restrictions.

The paper states that the unrestricted implication from integer similarity to strong shift equivalence remains unresolved in dimension two, while it is known to fail for certain larger matrices.

References

If $A$ and $B$ are $2 \times 2$ primitive nonnegative integer matrices similar over $\mathbb{Z}$, are they SSE ?

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Problem environment, Section 2, “Open problems”