Mahler's conjecture on prescribed rational powers of Liouville numbers

Let $I$ and $f_{a,b}$ be as in Theorem~. Let $A\subseteq\mathbb{Q}\setminus\{0\}$ satisfy $1\in A$ and the property that every non-zero integral multiple of each element of $A$ also belongs to $A$. Mahler's conjecture. There exist uncountably many $\zeta\in{\mathscr{L}\cap I}$ such that

$$f_{a,b}(\zeta)\in{\mathscr{L}}\quad\text{if }a/b\in A,$$

and

$$f_{a,b}(\zeta)\notin{\mathscr{L}}\quad\text{if }a/b\notin A.$$

The conjecture seeks to prescribe exactly which non-zero rational powers of a Liouville number remain Liouville, extending the preceding results for individual rational exponents. Its status is not established by the supplied text.