Geyer–Jarden conjecture on torsion phenomena over K(σ) for arbitrary finitely generated K

Prove that for any finitely generated field K over its prime field and any positive integer e, Statements (a)–(c) of Theorem 3.1(3) on torsion in Mordell–Weil groups of abelian varieties over K(σ) hold (namely: (a) for e = 1, A(K(σ))tor is infinite and A(K(σ))[ℓ] ≠ 0 for infinitely many primes ℓ; (b) for e ≥ 2, A(K(σ))tor is finite; and (c) for all e, for any prime ℓ, A(K(σ))[ℓ^∞] is finite).

Background

The paper collects known results validating these torsion statements in characteristic zero and for elliptic curves, and notes that they are conjectured in full generality by Geyer and Jarden. The positive characteristic cases (especially (a) and (b)) remain open for infinite K.