Polynomial recurrence characterization for Lucas atoms at $t=0$

Let $s\in\mathbb Z$. The sequence $(P_n(s,0))_{n\geq 1}$ is called polynomially recurrent if it satisfies a polynomial recurrence in $n$. Polynomial-recurrence conjecture. The sequence $(P_n(s,0))_{n\geq 1}$ is polynomially recurrent if and only if $s\in\{-1,0,1\}$. The preceding non-holonomicity result applies only when $t\neq 0$, while for $t=0$ one has $P_1(s,0)=1$ and $P_n(s,0)=s^{\varphi(n)}$ for $n\geq 2$; the conjecture asks exactly which integer values of $s$ yield polynomial recurrence.