I am reading the paper of Pink and Rutsche about the adelic openness of the image of Galois representations associated to Drinfeld modules, i.e.,
Pink, Rutsche, Adelic openness for Drinfeld modules in generic characteristic
In this paper, assume $F$ is a field of transcendental degree one over a finite field $\mathbb{F}_q$, and let $A$ be the ring of elements that are regular outside a fixed place $\infty$. And let $K/F$ be a finite extension. Now take $\phi: A\to K\{\tau\}$ to be a rank $r$ Drinfeld module of generic characteristic (in particular, the composite $a\mapsto \phi_a'$ from $A\to K$ is the natural embedding of $A$ into $K$.) Then they proved that the adelic representation
$$ \rho: {\rm Gal}(K^{sep}/K)\to \prod_{\mathfrak{p}\neq \infty} {\rm GL}_r(A_{\mathfrak{p}}) $$ has open image.
On page 887 of this paper, in the "Reduction steps" subsection, part (e), they claim that one can assume there exists a prime place $\mathfrak{p}$ of $F$, such that all places $\mathfrak P$ of $K$ lying above $\mathfrak{p}$ are unramified over it. My confusion is, what if $K/F$ is purely inseparable? In this case, it seems that this assumption cannot be guaranteed. Or, is there any well-known facts that lead to this reduction (for example, can $\phi$ be descended to a separable extension, or $\phi$ is isogenous to one defined over a separable extension?)
Any hint is appreciated.