Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.

**Question:** Is there a quadratic extension $L/K$ and a point $x\in C(L)$ which is non-degenerate in $J$, i.e. such that $\mathbb{Z}x$ is dense in $J$?

If $J$ has only finitely many abelian subvarieties, the answer is obviously yes. What about the case where $J$ is isogeneous to the square of an elliptic curve $E$? There are infinitely many elliptic curves on $J$ given by isogenies of $E$ and a priori, it seems possible to me though quite implausible that they could cover all quadratic points on $C$.

**Possible general statement:** I would expect that this is part of a much more general "unlikely intersection" statement for any hyperbolic curve $C$ embedded in an abelian variety and demanding $[L:K]<n$ such that points of degree $<n$ are dense in $C$; but I am mainly interested in the above.