Furstenberg's intersection problem for invariant Cantor sets

Let $A_2,A_3\subset[0,1]$ be closed sets invariant under multiplication by $2$ and $3$ modulo $1$, respectively. Suppose

$$\dim_{\mathrm{H}}A_2+\dim_{\mathrm{H}}A_3<1.$$

Furstenberg intersection conjecture. Every point of $A_2\cap A_3$ is rational. This is posed as a strengthening of known dimension and sparsity results for intersections of $\times2$- and $\times3$-invariant sets; the claim is not established in the paper.