The disk conjecture for subtree roots of trees

Let $T$) be a tree of order $n \geq 2$. A subtree root is a zero of the subtree polynomial of $T$. Define

$$D_n=\left\{z\in\mathbb{C}\colon\ |z|\leq 1+\sqrt[n-1]{n-1}\right\}.$$

Disk conjecture. The subtree roots of $T$ lie in $D_n$.

This strengthens the proved bound for all trees by making the radius depend on the order $n$; the authors verified the conjecture for all $n\leq 18$.