Obstructions to applying the Baker--Bilu method for determining integral points on curves

We prove that for every smooth projective integral curve $X$ of genus at least $2$ over $\mathbb{C}$, there exists $x\in X(\mathbb{C})$ such that no connected finite étale cover of $X-\{x\}$ admits a nonconstant morphism to ${\mathbb{G}}_m$. This has implications for the applicability of Baker’s method to determining integral points on curves.