Uniqueness conjecture for gravitating vortices on the projective line

Let $L$ be the line bundle and $\phi$ the vortex data on ${\mathbb P}^1$; let $\tau$ satisfy the inequality in equation, and let $\alpha>0$. The datum $\phi$ is polystable. Uniqueness conjecture. There exists a unique solution of the gravitating vortex equations on $({\mathbb P}^1,L,\phi)$ modulo automorphisms. This conjecture proposes existence and uniqueness for the Einstein--Bogomol'nyi equations on the projective line under the stated stability and parameter conditions; the surrounding discussion motivates it through the geometry of geodesics and the moduli space, but gives no resolution.