The Carlitz-exponential μ-transcendence dichotomy

Let $K_s$ and $\mathbb{K}_s$ be the fields in the Tate-algebra setting, and let $\mu$-transcendence and regular $\mu$-transcendence be understood in the sense defined in the source. For every nonzero $f\in\mathbb{K}_s$,

Dichotomy conjecture. Either $f$ or $\exp_C(f)$ is regularly $\mu$-transcendental over $K_s$.

The source says that this statement follows from the preceding functional conjecture for $n=1$, but supplies no proof of the functional conjecture or of this consequence.