Fix a number field $K$.

Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?

Does there exist an integer $g$ such that the rank of $J(K)$ is unbounded, where $J$ now ranges over the Jacobians of all smooth, projective, geometrically connected curves of genus $g$ over $K$?

I expect (perhaps naively) that the answer to 1. is "yes". Maybe one can write down an explicit family of superelliptic curves and use descent to show that their ranks are not bounded. Or else there may be a construction that, given an elliptic curve $E$ and integer $r$, produces a curve $C$ such that $\mathrm{Jac}(C)$ contains a factor $E^r$ up to isogeny. But I can't make either approach work.

On the other hand, I would be surprised if 2. were known, since it is so famously open in the case $g=1$.