Uniqueness conjecture for finite-radius Bradlow vortices on a flat disc

Let $\mathbb{D}^2$ be a flat disc of finite radius $R<\infty$, let $u$ solve the Bradlow equation, and impose the boundary condition $u(R)=0$. For vortex positions $z_i$, the axially symmetric solution is

$$u = \frac{R^2-r^2}{4} + \frac{N}{2}\log\frac{r^2}{R^2}.$$

Bradlow vortex uniqueness conjecture. The only solution satisfying the Bradlow equation on $\mathbb{D}^2$ with $u(R)=0$ is the axially symmetric solution above, with all vortex positions $z_i=0$ for every $i$.

The claim asserts uniqueness of the strictly boundary-compatible finite-disc solution, in contrast with solutions whose boundary discrepancy can be made arbitrarily small when $R\gg |z_i|$. The source provides no resolution, so the conjecture remains open.