  In this paper, we introduce the notion of $L^2$-subgroup rigid groups and
demonstrate that free groups are $L^2$-subgroup rigid. As a consequence, we
establish the equivalence between compressibility, inertness, strong inertness,
and $L^2$-independence for a finitely generated subgroup of a free group,
confirming a conjecture by Dicks and Ventura as well as the one by Antolin and
Jaikin-Zapirain.
