The reverse logarithmic Kodaira inequality and additivity

We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism $f:(X,E)\to (Y,D)$ of smooth projective reduced simple-normal-crossing pairs, with $\mathrm{Supp}(f^{\ast }D)\subseteq \mathrm{Supp}E$, such that X and every boundary stratum are smooth over $Y\setminus \mathrm{Supp}D$. Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.