Proper forcing extensions and Solovay models

We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every (Sigma) under tilde (1)(3) strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of L(R) under (Sigma) under tilde (1)(3) strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of L(R) under projective strongly-proper forcing notions is consistent relative to the existence of a (Sigma) under tilde (omega)-Mahlo cardinal. We also show that the consistency strength of the absoluteness of L(R) under forcing extensions with sigma-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.