A counterexample to Tachikawa's second conjecture

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over $k={\mathbb{F}}_2(q,H_1,H_2)$ with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.