Polynomial solvability conjecture for the formal asymptotic expansion

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ be a polynomial satisfying

$$\pi_0=0,\qquad \pi_1=\pi_2,$$

and set $p_0(x)=x$. Let $p_k$ denote polynomials whose coefficients are determined recursively by the formal expansion referred to in the source, and let the associated differential equation be the ODE for which that expansion is constructed. Polynomial solvability conjecture. For every $k\in\mathbb N_+$, there exist polynomials $p_k$ such that the formal series satisfies the ODE. Numerical experiments and symbolic computations support polynomial solvability for the first several values of $k$, but the claim for all $k$ remains open.