Interior conjecture for fixed points of finitely generated Galois subgroups on curves

Let $K$ be an infinite field, let $X$ be a curve over $K$, let $K^{\operatorname{sep}}$ be a separable closure of $K$, and let $G_K$ be the absolute Galois group of $K$. For $n\in\mathbb{N}$ and $(\sigma_1,\ldots,\sigma_n)\in G_K^n$, write $X(K^{\operatorname{sep}})^{\langle\sigma_1,\ldots,\sigma_n\rangle}$ for the points fixed by the subgroup generated by the $\sigma_i$. Interior conjecture. The set

$$\{(\sigma_1,\ldots,\sigma_n)\in G_K^n\colon |X(K^{\operatorname{sep}})^{\langle\sigma_1,\ldots,\sigma_n\rangle}|=\infty\}\subset G_K^n$$

has non-empty interior. The preceding corollary establishes this for hyperelliptic curves over infinite fields of characteristic different from $2$; the conjecture asks for the same conclusion for every curve over every infinite field.