Degree conjecture for general rank sequences

Let a recurrence have characteristic polynomial with $k$ distinct roots, without multiplicity, and let $N\subseteq\mathbb{Z}^k$ be the subgroup of relations between these roots. Write $d$ for the rank of the free part of the quotient group $\mathbb{Z}^k/N$. Degree conjecture. The associated general rank sequence is eventually given by a polynomial of degree $d-1$. This conjecture refines the broader claim that general rank sequences are eventually polynomial. The examples suggest that each independent relation lowers the degree, but the source supplies no proof or resolution status.