Becker's categorical rationality conjecture for $k$-regular power series

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f\in\mathpzc{k}[[z]]$. A rational function $h(z)\in\mathpzc{k}(z)$ is $k$-regular if it is a $k$-regular rational function, and a Mahler equation has polynomial coefficients when all its coefficient functions lie in $\mathpzc{k}[z]$. Becker's conjecture. The power series $f$ is $k$-regular if and only if there exists a nonzero $k$-regular rational function $h(z)\in\mathpzc{k}(z)$ such that $f(z)/h(z)$ satisfies a Mahler equation with polynomial coefficients. One implication is known by Becker; the converse remains open.