Zero-distribution conjecture for generated Taylor polynomials with nonpositive discriminant

Let $\{P_m(z)\}_{m\geq 0}$ be a sequence of functions of $z$ generated by

$$\sum_{m=0}^\infty P_m(z)t^m=\frac{1}{(at^2+bt+c)(1-tz)}.$$

Here $a,b,c\in\mathbb{R}\backslash\{0\}$ and $b^2-4ac\leq 0$. Let $t_1$ be the zero of $at^2+bt+c$ having smallest modulus, and let $\mathcal{Z}(P_m)$ denote the zero set of $P_m$. Zero-distribution conjecture.

$$\lim \mathcal{Z}(P_m)=\left\{z\in\mathbb{C}:|z|=\frac{1}{|t_1|}\right\}.$$

The claim concerns the limiting distribution of zeros for the generating-function sequence in the case where the quadratic has nonpositive discriminant. The paper's conclusions indicate that the complex-zero case remains to be handled, and the parser supplied no resolution evidence, so the status is left open.