The Mahler-measure-zero conjecture for finite-measure rank one maps

Let $(X,\mathcal{B},{\mathbb{P}},T)$ be a rank one map, where $(X,\mathcal{B},{\mathbb{P}})$ is a measure space and $T$ is the associated transformation. Assume that

$${\mathbb{P}}(X)<+\infty.$$

Mahler-measure-zero conjecture. The Mahler measure of the spectrum of $T$ is zero.

The paper proves the claim for subclasses of rank one maps, including a class with cutting parameter $m_k=\theta(k^\beta)$ for some $\beta\leq 1$, but leaves the general finite-measure case open. The conjecture is related to the conjecture that all rank one maps have singular spectrum.