Subsequential limit conjecture for generalized spiral polynomials

Let $p_k$ be the polynomial constructed from the generalized spiral points, and fix $z_0\in\mathbb{C}^*$. For a sequence $\lambda=\{\lambda_m\}$ with $\lambda_m\in[0,1[$, let $P_\lambda$ be a function with zeros at

$$\{n+\lambda_m+im:m,n\in\mathbb{Z}\}.$$

Generalized spiral limit conjecture. The rescaled functions $p_k\left(z_0+\frac{z}{\sqrt{k}}\right)$ have a subsequence converging to a normalization of $P_\lambda$ for some $\lambda$. This describes the expected local scaling limit of the spiral-point configuration; the source gives no proof or resolution.