Buff's conjecture on continuous dependence of invariant measures on the multiplier

Let $\mathbb{U}_X = \{\,\exp(2i\pi x)\,;x\in X\,\}$ and let $\mathcal{B}$ denote the set of Brjuno numbers. The function

$$\lambda\in\mathbb{D}\cup\mathbb{U}_\mathbb{Q}\cup\mathbb{U}_{\mathcal{B}}\mapsto \mu_\lambda$$

has a continuous extension to $\overline{\mathbb{D}}$ for the weak-$\ast$ topology on measures. Buff's conjecture. This predicts continuity, including at boundary multipliers, of the measures $\mu_\lambda$ associated with the family under consideration. The supplied text does not state whether the conjecture has been proved or disproved.