[Math] Is every abelian variety a subvariety of a Jacobian

Question

Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?

In these notes by William Stein this is stated without proof in remark 1.5.8; it is attributed to personal conversation with Brian Conrad. Unfortunately I was unable to locate or come up with any proof.

Answer 1

You can find a detailed proof here (theorem 1.2) in the case of principally polarized abelian varieties. One reduces to this case using the Zarhin's trick. The assumption of $k$ being infinite should not be necessary (see remark 1.3 in the paper)