Extensions of a dualizing complex by its ring: commutative versions of a conjecture of Tachikawa

Let (R,m,k) be a commutative noetherian local ring with dualizing complex DR, normalized by ExtRdepth(R)(k,DR)≅k. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) k-algebras of finite rank, we conjecture that if ExtRn(DR,R)=0 for all n>0, then R is Gorenstein, and prove this in several significant cases.