Extension of the polynomial strong-shift-equivalence construction

Determine whether the strong-shift-equivalence constructions for the polynomial matrix pairs $C_k$ and $D_k$ in Theorem 1 can be extended whenever a strong shift equivalence exists between two $2\times2$ matrices, or whether additional conditions are necessary.

Background

Theorem 1 constructs parameter-independent strong shift equivalences over the semiring of nonnegative polynomials for particular families of two-by-two matrices, including the cases $k=2,3,4$.

The conclusion explicitly asks whether this type of generalization is available for every strong-shift-equivalent pair of two-by-two matrices or whether the construction requires further hypotheses.