Nakamura - Caruana (FIDE Grand Prix Paris) : Can you spot the tactic?

In this position, GM Caruana (Black) made an enormous mistake. Only one move does not lose for Black. Can you find which one, and why the other moves fail?
Spoiler: Highlight to view
Caruana played 14... hxg6?? and quickly lost after 15.Bxd4 Qxd4 (15... Bxd4 16. Qh6) 16. Qe1 Qxd1 17.Nxd1  Instead, 14...fxg6 allows to capture back in d4 with the bishop, as Qh6 will be met by e5.