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.