The irrational multiplicatively independent Mahler Hahn-series conjecture

Let $\alpha$ and $\beta$ be irrational and multiplicatively independent. Let $F(x)$ be a Hahn series over $\mathbb{R}$ that is both $\alpha$-Mahler and $\beta$-Mahler. Irrational Mahler Hahn-series conjecture. Then $F(x)\in K(x)$. This asks whether the paper's rationality result for algebraically independent irrational parameters extends to all multiplicatively independent irrational parameters. The source presents this as an open question for further investigation.