Complex-parameter hypergeometric lemniscate conjecture

Let $\alpha=\eta+izeta$ with $\eta>0$ and $\zeta\neq0$, and define

$$p_n(z)={}_{2}\operatorname{F}_{1}\left[ \begin{array}{c} -n, \alpha n + 1, \\ \alpha n+2 \end{array}; z\right].$$

Complex-parameter clustering conjecture. The zeros of $p_n$ asymptotically cluster on the loop of the lemniscate

$$|z^{\eta}(1-z)|e^{-\zeta\operatorname{Arg} z}=\frac{|\alpha|^{\eta}}{|\alpha+1|^{\eta+1}}e^{\zeta\operatorname{Arg}\left(\frac{\alpha}{\alpha+1}\right)},$$

for $\operatorname{Re}(z)>\frac{\eta}{\eta+1}$. This is a more explicit complex-parameter version of the preceding level-curve prediction; the source gives no resolution, so its asymptotic clustering claim remains open.