Furstenberg's conjecture on projections of ×2- and ×3-invariant sets

Let $T_m:[0,1]\to[0,1]$ be the map $x\mapsto mx\bmod 1$, and let $\Pi_{2,1}$ be the space of orthogonal projections from $\mathbb{R}^2$ to one-dimensional subspaces. Let $\pi_x,\pi_y\in\Pi_{2,1}$ be the coordinate projections onto the axes. Suppose that $X,Y\subseteq[0,1]$ are closed, with $X$ invariant under $T_2$ and $Y$ invariant under $T_3$. Furstenberg's conjecture. For every $\pi\in\Pi_{2,1}\setminus\{\pi_x,\pi_y\}$,

$$\dim\pi(X\times Y)=\min\{1,\dim(X\times Y)\}.$$

The coordinate projections are excluded because they map $X\times Y$ to $X$ or $Y$, respectively, and can therefore exhibit an evident dimension drop. The conjecture concerns exceptional projections of dynamically defined product sets and is equivalent here to the corresponding sumset assertion; the paper proves it.