Furstenberg's dimension conjecture for multiplicatively independent orbits

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let $x\in[0,1)$ be a real number. Define the forward orbit

\mathcal O_q(x):=\left\\{x,T_q(x),T_q^2(x),\ldots\right\\},

where $T_q$ is the map on $\mathbb R/\mathbb Z$ given by $x\mapsto qx$, and let $\dim_H$ denote Hausdorff dimension. Furstenberg's conjecture. Unless $x$ is rational,

$$\dim_H \overline{\mathcal O_p(x)}+\dim_H \overline{\mathcal O_q(x)}\geq 1.$$

The conjecture expresses the expected trade-off between the complexity of expansions of an irrational real number in two multiplicatively independent bases. Shmerkin and Wu proved that the exceptional set has Hausdorff dimension zero, but the conjecture remains open in general.