Disk conjecture for the roots of the descent polynomial

Let $I$ be a finite subset of the positive integers, let $m$ be the maximal element of $I\cup\{0\}$, and let $d(I,z)$ be the descent polynomial associated with $I$. For a root $z_0\in\mathbb C$ of this polynomial, consider the disk whose diameter has endpoints $-1$ and $m$.

Disk conjecture. If

$$d(I,z_0)=0,$$

then

$$\left|z_0-\frac{m-1}{2}\right|\le\frac{m+1}{2}.$$

This conjecture is motivated by numerical computations for $m\le 13$, the case $|I|=1$, and a result concerning real roots. It simultaneously implies the two previously discussed bounds, but the source does not state that it has been resolved.