  It is a well-known result of Auslander and Reiten that contravariant
finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely
generated modules of finite projective dimension) over an Artin algebra is a
sufficient condition for validity of finitistic dimension conjectures.
Motivated by the fact that finitistic dimensions of an algebra can
alternatively be computed by Gorenstein projective dimension, in this work we
examine the Gorenstein counterpart of Auslander--Reiten condition, namely
contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of
finitely generated modules of finite Gorenstein projective dimension), and its
relation to validity of finitistic dimension conjectures. It is proved that
contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$
implies validity of the second finitistic dimension conjecture over left
artinian rings. In the more special setting of Artin algebras, however, it is
proved that the Auslander--Reiten sufficient condition and its Gorenstein
counterpart are virtually equivalent in the sense that contravariant finiteness
of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariant
finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin
algebra, and the converse holds for Artin algebras over which the class
$\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective
modules) is contravariantly finite.
