In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for class forcing. The proof uses Jensen Coding. Is this still true in ZFC${^-}$? I am particularly interested whether generic $\mathbf{\Sigma_3^1}$-absoluteness might be consistent for class forcings over models of Second Order Arithmetic (resp. ZFC$^-+$V=HC).