Transcendence conjecture for twisted special $L$-values over finite extensions

Let $\mathbf{A}=\mathbb{F}_q[t]$, $A=\mathbb{F}_q[\theta]$, and $K=\operatorname{Frac}(A)=\mathbb{F}_q(\theta)$. Let $E$ be a finite extension of $K$ and let $G_E=\operatorname{Gal}(E^{\mathrm{sep}}/E)$. Let $\varphi$ be a Drinfeld module over $E$, and let $\rho:G_E\to\operatorname{GL}_n(\overline{\mathbb{F}}_q)$ be an Artin representation. Transcendence conjecture. For every non-negative integer $k$, the twisted special value $L(\varphi^\vee,\rho,k)$ is transcendental over $\overline{E}$. This extends the theorem proved in the paper for $E=K$ to Drinfeld modules and Artin representations defined over arbitrary finite extensions of $K$.