  In this paper, we consider a depth inequality of Auslander which holds for
finitely generated Tor-rigid modules over commutative Noetherian local rings.
We raise the question of whether such a depth inequality can be extended for
$n$-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules
that are generically free. Furthermore, in the appendix, we use Dao's eta
function and determine new classes of Tor-rigid modules over hypersurfaces that
are quotient of unramified regular local rings.
