Loxton and Van der Poorten's conjecture for Mahler functions

Let $k$ and $l$ be multiplicatively independent positive integers. Let $F\in K[[x]]$ be a power series over a number field $K$ that is both $k$-Mahler and $l$-Mahler. Loxton and Van der Poorten's conjecture. Then $F$ is rational. This is a Mahler-function analogue of Cobham's theorem, asserting that simultaneous multiplicative independence forces a power series to be rational. The conjecture was resolved in 2013 by Bell and Adamczewski, who proved the more general theorem over any field of characteristic $0$.