Convergence conjecture for the formal asymptotic expansion

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and set $p_0(x)=x$. Consider the formal expansion with recursively defined polynomial coefficients $p_k$, and let the convergence domain be the one specified by the source's convergence lemma. Convergence conjecture. Under these conditions, the hypotheses of the convergence lemma hold, and the formal expansion is an actually convergent series throughout the domain specified by that lemma. The claim is presented as a consequence suggested by numerical evidence, but the necessary coefficient bounds and convergence remain unproved in the supplied text.