Furstenberg's ×(2,3) conjecture for invariant measures

Let $\mathbb{T}$ be the unit circle, let $\operatorname{Prob}(\mathbb{T};2,3)$ denote the set of measures invariant under the maps $\omega\mapsto\omega^2$ and $\omega\mapsto\omega^3$, let ergodic mean an extreme point of this convex set, and let $\lambda$ be Lebesgue probability measure on $\mathbb{T}$. Furstenberg's ×(2,3) conjecture. If $\mu\in\operatorname{Prob}(\mathbb{T};2,3)$ is ergodic, then either $\mu=\lambda$ or $\operatorname{supp}(\mu)$ is a finite set. This is a central problem in ergodic theory, with connections to homogeneous dynamics and number theory; the source gives no evidence of resolution.