Nonnegative-coefficient conjecture for the formal asymptotic polynomials

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and let $p_0(x)=x$. Assume that the formal expansion has polynomial coefficients $p_k$ as in the polynomial solvability conjecture. Nonnegative-coefficient conjecture. If

$$\pi_j\geq 0\qquad (j=0,\dots,d),$$

then every coefficient of every resulting polynomial $p_k$ is nonnegative. Numerical experiments support the claim, including the cases $P(t)=t^d$ for $d=3,4,6$, but no proof is given and the conjecture remains open.