Larsen’s conjecture: infinite rank over K(σ) for each σ

Prove that if K is an infinite finitely generated field and e ≥ 1, then for every σ in G_K^e and every abelian variety A of positive dimension over K(σ), the Mordell–Weil group A(K(σ)) has rank ℵ₀.

Background

Theorem 3.1(1) asserts that for almost all σ in G_K^e the rank is ℵ₀. Larsen conjectured that the same holds for every σ (not just almost all). Im and Larsen proved the conjecture when e=1 and char(K)≠2, but the general case is open.