Freeness modulo torsion over finite extensions of K[σ] when e = 1

Determine whether, when e = 1 and K is finitely generated over Q, for almost all σ ∈ G_K and every finite extension L of K[σ], the quotient A(L)/A(L)_{tor} is a free Z-module of rank ℵ₀ for every semiabelian variety A of positive dimension over L.

Background

Theorem 4.1 proves that for e ≥ 2, almost all σ, and any finite extension L of K[σ], the group A(L)/A(L)_{tor} is free of rank ℵ₀. The authors note that this freeness result is unknown when e = 1 due to the presence of large torsion in that case.