Gau–Wang–Wu conjecture about nilpotent partial isometries

Let $A\in M_n$ be a nilpotent partial isometry, and suppose that $A$ is unitarily irreducible, meaning that $A$ has no nontrivial reducing subspace. Its numerical range is

$$W(A)=\{\langle Ax,x\rangle:x\in\mathcal H,\ \lVert x\rVert=1\}.$$

The matrix $A$ is rotationally invariant when, for every $\theta\in\mathbb R$, the matrices $A$ and $e^{i\theta}A$ are unitarily similar.

Gau–Wang–Wu conjecture. If the numerical range $W(A)$ is a circular disc centered at the origin, then $A$ is rotationally invariant.

This conjecture concerns the distinction between circular symmetry of the numerical range and the stronger structural condition of unitary similarity to every scalar rotation. The source presents it as one of three conjectures on partial isometries; its resolution is not established in the supplied text.