Furstenberg's weak-star convergence conjecture for times $p$-invariant measures

Let $p,q\ge 2$ be fixed multiplicatively independent integers, let ${\mathbb T}={\mathbb R}/{\mathbb Z}$ be the circle group, and let $T_n(x)=nx\mod 1$. A Borel probability measure on ${\mathbb T}$ is continuous if it has no atom and is $T_p$-invariant when $\mu=T_p\mu$. Write $w^*$ for the weak-star topology on measures.

Furstenberg's weak-star convergence conjecture. If $\mu$ is a continuous Borel probability measure on ${\mathbb T}$ that is $T_p$-invariant, then $T_q^n\mu$ converges in the weak-star topology to normalized Lebesgue measure $\operatorname{Leb}$.

This is the natural measure analogue of the Hausdorff convergence conjecture for invariant sets. The source identifies it as Furstenberg's Conjecture (C3) and states that it remains conjectural.