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

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 $\mathbb{T}$.