Eventual polynomiality of general rank sequences

Let $s=(s(n))_{n\geq 0}$ be a constant-recursive sequence. A general rank sequence is the rank sequence determined by a recurrence and its associated general exponential-polynomial data, while a particular rank sequence is the sequence $\left(\operatorname{rank}(s(n)^M)\right)_{M\geq 1}$ for a specific sequence. Polynomiality conjecture. Every general rank sequence is eventually polynomial, and every particular rank sequence is eventually pseudo-polynomial. The paper gives examples of eventual polynomial behavior and discusses coefficient cancellation in particular rank sequences. No resolution is supplied.