I found this pretty fascinating; the two really are tremendously close to joining together, see [1] (closest point of Hominy creek) and [2] (closest point of the Pigeon River). Hominy sits slightly above the Pigeon at this closest point, so the capture would find flow reversing from dumping into the French Broad to dumping into the Pigeon.

[1] - 35.531518, -82.832689

[2] - 35.52846, -82.840061