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Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein localring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of smallcolength, then every totally reflexive $R$-module is free. Indeed, the secondsyzygy of the canonical module of $R$ has a direct summand $T$ which is a testmodule for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$,for some finitely generated $R$-module $N$, then $N$ is free.

Totally reflexive modules over rings that are close to Gorenstein