  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 $\mathrm{Tor}_+^R(T,N)=0$,
for some finitely generated $R$-module $N$, then $N$ is free.
