Modules of G-dimension zero over local rings of depth two

Let $R$ be a commutative noetherian local ring. Denote by $\mod R$ the category of finitely generated $R$-modules, and by ${\mathcal G} (R)$ the full subcategory of $\mod R$ consisting of all $R$-modules of G-dimension zero. Suppose that $R$ is henselian and non-Gorenstein, and that there is a non-free $R$-module in ${\mathcal G} (R)$. Then it is known that ${\mathcal G} (R)$ is not contravariantly finite in $\mod R$ if $R$ has depth at most one. In this paper, we prove that the same statement holds if $R$ has depth two.