The Baker–Bilu obstruction conjecture for integral points on curves

Let $X$ be a nice curve, meaning a smooth, projective, geometrically integral curve, of genus at least $2$ over $\overline{\mathbb{Q}}$. For a point $x\in X(\overline{\mathbb{Q}})$, write $X-\{x\}$ for the resulting affine curve. Baker–Bilu obstruction conjecture. For every such $X$, there exists $x\in X(\overline{\mathbb{Q}})$ such that $X-\{x\}$ has no connected finite étale cover admitting even a single nonconstant morphism to $\mathbb{G}_m$. This is a stronger proposed negative answer to the question of whether every smooth integral affine curve of genus at least $2$ over $\overline{\mathbb{Q}}$ admits a connected finite étale cover with a nondegenerate morphism to $\mathbb{G}_m\times\mathbb{G}_m$. If true, it would obstruct applying the Baker–Bilu method to determining integral points on curves.