The annulus 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 $T$. Consider the annulus

$$\left\{z\in\mathbb{C}\colon\ \tfrac{1}{2}\leq \left|z+\tfrac{1}{2}\right|\leq \tfrac{1}{2}+\sqrt[n-1]{n-1}\right\}.$$

Annulus conjecture. The subtree roots of $T$ are contained in this annulus.

This is presented as a strengthening of the disk conjecture: it gives both an inner exclusion region and an order-dependent outer bound. The conjecture remains open.