Becker's naive precalmness conjecture

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ satisfy the Mahler equation

$$f(z)=\sum_{i=1}^{n}c_i(z)f\left(z^{k^i}\right).$$

A sequence of coefficients is precalm in the sense defined in the paper. Becker's naive conjecture. The power series $f$ is regular if and only if the sequence $(c_i)$ is precalm. This is proposed as a naive version after the paper's regularity criteria and remains open.