The conjecture that transcendental elements of the Mahler-value set are S-numbers

Let $\mathbf M$ be the set of values under consideration in the paper. For a complex number $\xi$ and a positive integer $d$, define $w_d(\xi)$ as the supremum of the real numbers $w$ for which $0<|P(\xi)|<H(P)^{-w}$ holds for infinitely many polynomials $P\in\mathbb Z[X]$ of degree at most $d$, and set

$$w(\xi):=\limsup_{d\to\infty}\frac{w_d(\xi)}{d}.$$

A complex number is an S-number when $0<w(\xi)<\infty$. The S-number conjecture for $\mathbf M$. Every transcendental element of $\mathbf M$ is an S-number. The conjecture is motivated by the expectation that transcendental elements of $\mathbf M$ should not be Liouville or U-numbers. The source states that the conjecture is open in general: the order-one case is known by Galochkin, while subsequent results only show that the number is either an S-number or a T-number.