Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?

Yes. With the known implications it follows that the Auslander-Reiten conjecture, the generalized Nakayama conjecture, the Auslander-Gorenstein conjecture, the Nakayama conjecture, the Gorenstein-projective conjecture and Tachikawa's second conjecture are all equivalent, and that Tachikawa's second implies his first. None of them is thereby proved.

Yes. With the known implications it follows that the Auslander-Reiten conjecture, the generalized Nakayama conjecture, the Auslander-Gorenstein conjecture, the Nakayama conjecture, the Gorenstein-projective conjecture and Tachikawa's second conjecture are all equivalent, and that Tachikawa's second implies his first. None of them is thereby proved.