Larsen’s conjecture on rank over K(σ) for every σ

Prove that for a finitely generated field K over its prime field, a positive integer e, any e-tuple σ ∈ G_K^e, and any abelian variety A of positive dimension over K(σ), the Mordell–Weil group A(K(σ)) has countably infinite rank.

Background

Theorem 3.1(1) asserts that for almost all σ ∈ G_K^e, the rank of A over K(σ) (and K[σ]) is countably infinite. Larsen conjectured that the “almost all” qualifier can be removed for the K(σ) case, i.e., that the conclusion holds for every σ. Partial results are known in special cases (e.g., e = 1 and char(K) ≠ 2).