On the section conjecture for the toric fundamental group

The toric fundamental group is the Tannaka dual of a category of vector bundles which become direct sums of line bundles on a finite étale cover. It is an extension of the étale fundamental group scheme by a projective limit of tori.
Grothendieck's section conjecture for the étale fundamental group implies the analogous statement for the toric fundamental group. We call this the toric section conjecture. We prove that a resolution of the toric section conjecture would reduce the original one to particular cases about which more is known, mainly due to J. Stix.
We prove that abelian varieties over p-adic fields satisfy the toric section conjecture, and give strong evidence that it holds for hyperbolic curves over p-adic fields, too.