General projection conjecture for irrational $(\times m,\times n)$-invariant measures

Let $T_{m,n}:\mathbb{T}^2\to\mathbb{T}^2$ denote the coordinatewise action of $T_m$ and $T_n$, let $\Pi_{2,1}$ be the set of orthogonal projections from $\mathbb{R}^2$ to $\mathbb{R}$, and let $\pi_1,\pi_2$ be the coordinate projections. General projection conjecture. Suppose that $\log m/\log n$ is irrational and that $\mu$ is a $T_{m,n}$-invariant measure. Then, for every $\pi\in\Pi_{2,1}\setminus\{\pi_1,\pi_2\}$,

$$\dim \pi\mu=\min\{1,\dim\mu\}.$$

This is posed in the prospects section as a natural direction beyond the Bernoulli-measure result proved earlier in the paper; its general case remains open according to the supplied context.