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on Furstenberg's times $p$, times $q$ conjecture

The statement

Let $p$ and $q$ be natural numbers with $p \geq 2$ and $q \geq 2$ . Two integers $p, q \geq 2$ are called multiplicatively independent if $\log p / \log q$ is irrational. Let $\mathbb{T}$ denote the unit additive circle. For each natural number $n$ , define the map $T_n : \mathbb{T} \to \mathbb{T}$ by $T_n(x) = nx \pmod{1}$ . Let $\mu$ be a Borel probability measure on $\mathbb{T}$ that is atomless, meaning that for every measurable set $A$ with $\mu(A) > 0$ , there exists a measurable subset $B \subseteq A$ such that $0 < \mu(B) < \mu(A)$ . Suppose that $\mu$ is measure-preserving for both $T_p$ and $T_q$ , meaning that $\mu(T_p^{-1}(E)) = \mu(E)$ and $\mu(T_q^{-1}(E)) = \mu(E)$ for every Borel set $E \subseteq \mathbb{T}$ . Prove that if $p$ and $q$ are multiplicatively independent, then $\mu$ equals the Lebesgue measure on $\mathbb{T}$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.