Tachikawa's second conjecture for self-injective algebras

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional symmetric algebra $A$ and a nonprojective finite-dimensional module $M$ with $\mathrm{Ext}^i_A(M,M)=0$ for all $i>0$, so Tachikawa's second conjecture fails, and $M$ is also a symmetric counterexample to the Auslander-Reiten conjecture. The conclusions persist under every field extension. It does NOT address Tachikawa's first conjecture separately, nor give characteristic-zero examples.

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional symmetric algebra $A$ and a nonprojective finite-dimensional module $M$ with $\mathrm{Ext}^i_A(M,M)=0$ for all $i>0$, so Tachikawa's second conjecture fails, and $M$ is also a symmetric counterexample to the Auslander-Reiten conjecture. The conclusions persist under every field extension. It does NOT address Tachikawa's first conjecture separately, nor give characteristic-zero examples.