Unrestricted calm-coefficient criterion for regular Mahler power series

Let $k\geq 2$, let $\mathpzc{k}$ be the coefficient field, and let $f(z)\in\mathpzc{k}[[z]]$ be a Mahler power series. A sequence of rational functions is calm in the sense defined in the paper. The unrestricted calmness conjecture. The power series $f$ is regular if and only if it satisfies a Mahler equation

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

whose coefficient sequence $(c_i)_{i=1}^{n}$ is calm. The paper proves this when the coefficients of the minimal-order Mahler equation have no poles at roots of unity; removing that assumption is the open part.