On forcing projective generic absoluteness from strong cardinals

W.H. Woodin showed that if ${\kappa }_1<\cdots <{\kappa }_n$ are strong cardinals then two-step ${\mathbf{\Sigma }}_{n+3}^1$ generic absoluteness holds after collapsing $2^{2^{{\kappa }_n}}$ to be countable. We show that this number can be reduced to $2^{{\kappa }_n}$, and to ${\kappa }_n^{+}$ in the case $n=1$, but cannot be further reduced to ${\kappa }_n$.