Uniqueness conjecture for the Angelesco equilibrium measure

In the Angelesco case, let $\vec\lambda=(\lambda_1,\lambda_2)$ be the equilibrium measure of the extremal vector compact, and set $\mu=\lambda_1+\lambda_2$. Let $\lambda(\mu,\mathcal F_i)$ be the equilibrium measures associated with the admissible-cut classes $\mathcal F_i$, so that

$$\mu\geq\lambda_1=\lambda(\mu,\mathcal F_1),\qquad \mu\geq\lambda_2=\lambda(\mu,\mathcal F_2).$$

Angelesco uniqueness conjecture. The measure $\mu=\lambda_1+\lambda_2$ is the only positive Borel measure in the plane satisfying these inequalities and $\mu(\mathbb C)=2$. This uniqueness would provide the potential-theoretic inverse step needed for the general Angelesco zero-distribution result; the source does not state that it has been proved.