Artin's conjecture for supercharacter theories

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $(X,K)$ be a supercharacter theory of $G$, and let $\operatorname{Sup}(G)$ be the set of supercharacters for this theory. Let $L(s,\sigma,K/F)$ denote the Artin $L$-function attached to a supercharacter $\sigma$. Supercharacter-theoretic Artin conjecture. For every nontrivial $\sigma\in\operatorname{Sup}(G)$, the function

$$L(s,\sigma,K/F)$$

extends to an entire function. The paper presents this as a supercharacter-theoretic variant of Artin's conjecture; the maximal supercharacter theory gives a known example, while the assertion for general supercharacter theories is posed as a conjecture.