Furstenberg's intersection conjecture for multiplicatively independent dynamics

Let $p,q\ge 2$ be integers with $p\nsim q$, where $a\nsim b$ means that $\log a/\log b\notin\mathbb Q$. Let $T_m:[0,1]\rightarrow[0,1]$ be the map $x\mapsto mx\mod 1$, and let $\dim_{\rm H}$ denote Hausdorff dimension. A closed set is $T_m$-invariant when it is invariant under $T_m$. Furstenberg's intersection conjecture. If $A_p,B_q\subset[0,1]$ are closed and invariant under $T_p$ and $T_q$, respectively, then for all real numbers $u$ and $v$,

$$\dim_{\rm H}((uA_p+v)\cap B_q)\le \max\{0,\dim_{\rm H}A_p+\dim_{\rm H}B_q-1\}.$$

The paper states that this conjecture is proved there, so it is a solved form of Furstenberg's transversality problem for the two multiplicatively independent maps.