Quasi-polynomiality conjecture for profiles of bounded-signature structures

Let $R$ be a relational structure. Its profile $\varphi_R(n)$ counts the isomorphism types of substructures induced on the $n$-element subsets of $R$. Say that $R$ has bounded signature when the arities of the relations in its signature are uniformly bounded, and let the kernel of $R$ denote the set of elements that do not belong to any infinite monomorphic block. Assume either that $R$ has bounded signature or that its kernel is finite.

Quasi-polynomiality conjecture. If $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is eventually a quasi-polynomial.

The kernel condition is motivated by structural results relating polynomially bounded profiles to finite kernels, while examples show that finite generation of the associated age algebra need not follow from polynomial growth. Eventual quasi-polynomiality would imply strong polynomial growth, but the conjecture is presented as unresolved.