The Hausdorff-measure conjecture for intersections of invariant sets

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

$$\dim_{\mathrm{H}}A_2+\dim_{\mathrm{H}}A_3=s\in(0,1].$$

For $g(x)=\exp(-(-\log x)^s)$, Hausdorff-measure conjecture. One has

$$\mathcal{H}^g(A_2\cap A_3)=0.$$

The paper proves the analogous vanishing result in the more restrictive range $s<1/27$ with exponent $27s$ in place of $s$; the stated strengthening remains open.