The rational-base equidistribution conjecture for iterated ceiling maps

Let $p>q\geq 1$ be coprime integers and define $T_{p/q}(x)=\left\lceil\frac{p}{q}x\right\rceil$. The rational-base equidistribution conjecture. For every $n\in\mathbb N_{>0}$ and every nonnegative integer $k$, the sequence

$$(T_{p/q}^l(n))_{l\in\mathbb N}$$

is equidistributed in the residue classes modulo $q^k$. The paper states that this conjecture is equivalent to the normality conjecture for minimal words and hence is likewise open in general.