Optimal Gegenbauer measure for logarithmic energy

Let $n\geq 1$ be fixed. Consider the problem of choosing a measure on the unit interval to minimize the expected logarithmic energy of the associated point process. Write $d\sigma=w^\lambda dx$, where $\lambda\in(-1/2,\infty)$.

Optimal-measure claim. For any fixed $n\geq 1$, the optimal measure is $d\sigma=w^\lambda dx$ for some $\lambda\in(-1/2,\infty)$. The value of $\lambda$ may depend on $n$.

This is presented as the authors' answer to Question I) in their discussion of Gegenbauer determinantal point processes and the optimal measure problem. The supplied text does not state whether the claim has been proved or remains open.