Larsen’s question on the rank of the free summand over k(A(k)_tor)

Determine whether, for a general abelian variety A over a finitely generated field k of characteristic zero, the free Z-module summand M in the decomposition A(K) = M ⊕ (Q/Z)^{2g} with K = k(A(k)_{tor}) has infinite rank.

Background

For K = k(A(k)_{tor}), Larsen proved that A(K) decomposes as a direct sum of a free module M and a divisible part (Q/Z)^{2g}, and showed rank(M)=0 for elliptic curves. He asked whether rank(M) is infinite for general abelian varieties, a problem that the present paper notes remains open.