Little shift equivalence conjecture for n=2

Prove or disprove that every nonnegative integer matrix with exactly one nonzero eigenvalue, equal to $2$ with multiplicity one, is strong shift equivalent to the one-by-one matrix $\begin{pmatrix}2\end{pmatrix}$.

Background

The little shift equivalence conjecture specializes the general strong-shift-equivalence problem by fixing the target matrix to (2)(2). It asks whether the specified spectral condition is sufficient for strong shift equivalence to that matrix.

Ashley’s eight-by-eight matrix is presented as a concrete instance of this conjecture. The paper resolves that particular instance, but the general conjecture remains an open problem.

References

If $A$ is a nonnegative integer matrix with a single nonzero eigenvalue $2$ of multiplicity 1, is $A$ SSE to the matrix $\begin{pmatrix} 2 \end{pmatrix}$ ?

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