Eventual quasi-polynomiality for profiles with bounded signature or finite kernel

Let $R$ be a relational structure with profile $\varphi_R$. Assume that $R$ has bounded signature or finite kernel, and that $\varphi_R(n)$ is bounded by some polynomial in $n$. Eventual quasi-polynomiality conjecture. The profile $\varphi_R$ is eventually a quasi-polynomial. Profiles are Hilbert functions of associated age algebras when finite-valued, and finitely generated graded commutative algebras have eventually quasi-polynomial Hilbert functions. The paper presents partial answers to related finite-generation and Cohen–Macaulay questions, but this broader conjecture remains open.