The Tachikawa conjecture states that $Ext^i(M,M) \neq 0$ for some $i \geq 1$ for every non-projective module $M$ over a selfinjective finite dimensional algebra.
In theorem 4.6. of http://maths.nju.edu.cn/~huangzy/When%20are%20torsionless%20modules%20projective.pdf , the authors prove something (much stronger!) which implies the Tachikawa conjecture in the commutative case, which would be a sensational result in my opinion.
Is the proof really true/without gaps? I couldnt understand everything and the authors never replied. Of course it might be extremely hard to give counterexamples to theorem 4.6. there but maybe gaps could be pointed out?