Hardy–Petrovitch–Hutchinson conjecture on iteration of hyperbolic polynomials

Fix $q\in(0,1)$, and let $\widetilde q$ be the distinguished parameter for which the partial theta function belongs to the Laguerre–Pólya class. Let $f_1=(x+1)Q(x)$, where $Q(x)$ is a hyperbolic polynomial whose roots are all smaller than $-1$, and let $\{f_j(x)\}$ be the polynomial sequence produced by the iteration scheme in the paper. Let $u(q)$ be the negative root of $\Psi(q,u)=0$ having minimal absolute value.

Iteration-convergence conjecture. If $0<q\leq\widetilde q$, then $\{f_j(x)\}$ converges uniformly on $[-1,0]$, together with all derivatives, to

$$\Psi(q,-u(q)x).$$

This conjecture concerns convergence of the polynomial iteration to the canonical fixed point represented by the smallest-magnitude negative zero of the partial theta function. The source presents it as suggested by computer experiments and the limiting theorem; its general validity remains open.