Totally reflexive modules over rings that are close to Gorenstein

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Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor+ʳ(T,N)=0, for some finitely generated R-module N, then N is free.