Edrei–Saff–Varga modified width conjecture

Let $f$ be an entire function of positive, finite order $\lambda$, and let $p_n(z)$ denote its $n^\text{th}$ partial sum. Edrei–Saff–Varga's modified width conjecture. There exist an infinite sequence of positive integers $N$ and finitely many exceptional arguments $\theta_1,\theta_2,\ldots,\theta_q$ such that the following hold: for every argument $\theta\ne\theta_j$, $j=1,2,\ldots,q$, there is a positive sequence $\rho_n$, $n\in N$, with $\rho_n\to\infty$ and $\rho_n=O(n^{2/\lambda})$, for which, for every fixed $\epsilon>0$, the number of zeros of $p_n(z)$ in

$$\left|z-\rho_ne^{i\theta}\right|\leq\rho_nn^{-1+\epsilon}$$

tends to infinity as $n\to\infty$, $n\in N$; and for every exceptional argument $\theta_j$, there is an integer $m\geq2$ and a positive sequence $\rho_n$, $n\in N$, with $\rho_n\to\infty$ and $\rho_n=O(n^{2/(\lambda m)})$, for which, for every fixed $\epsilon>0$, the number of zeros of $p_n(z)$ in

$$\left|z-\rho_ne^{i\theta_j}\right|\leq\rho_nn^{-1/m+\epsilon}$$

tends to infinity as $n\to\infty$, $n\in N$. The modified conjecture describes dense clustering of zeros in all but finitely many directions, with slower concentration allowed in exceptional directions associated with maximal exponential growth; its resolution is not specified in the source.