Tachikawa's commutative conjecture on canonical modules

Let $(R,\mathfrak m,k)$ be a Noetherian local ring with a canonical module $K_R$. Tachikawa's conjecture. If

$$\operatorname{Ext}^i_R(K_R,R)=0$$

for all $i>0$, then $R$ is Gorenstein. This is the commutative version of Tachikawa's first conjecture concerning the vanishing of Ext modules; the supplied source does not state whether the conjecture has been resolved.