Saff–Varga width conjecture for zeros of partial sums

Let $f$ be an entire function of positive finite order $\lambda>\tau$, and let $p_n(z)$ denote its $n^\text{th}$ partial sum. For fixed positive constants $K$ and $x_0$, define the parabolic region

$$S_0(\tau)=\left\{z=x+iy:|y|\leq Kx^{1-\tau/2},\ x\geq x_0\right\},$$

and, for each argument $\theta$, let

$$S_\theta(\tau)=e^{i\theta}S_0(\tau).$$

Saff–Varga's width conjecture. There exists an infinite sequence of positive integers $N$ such that no region $S_\theta(\tau)$ is devoid of all zeros of all partial sums $p_n(z)$ with $n\in N$. This asserts that zero-free regions for partial sums cannot be too wide, with the allowable width governed by the order of $f$; the conjecture is attributed to Saff and Varga and its resolution is not specified in the source.