The section conjecture for the toric fundamental group over $p$-adic fields

The toric fundamental group is the Tannaka dual of a category of vector bundles which become direct sums of line bundles on a finite \'etale cover. It is an extension of the \'etale fundamental group scheme by a projective limit of tori. Grothendieck's section conjecture for the \'etale 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.