Invariance of plurigenera under smooth projective deformations

Let $\pi:X\to S$ be a smooth projective deformation, with fibres $X_s=\pi^{-1}(s)$ and plurigenera given by

$$P_m(X_s)=h^0(X_s,K_{X_s}^{\otimes m}).$$

Invariance of plurigenera conjecture. The plurigenera is invariant under smooth projective deformations; equivalently, for every integer $m\geq 1$, $P_m(X_s)$ is independent of $s\in S$.

The theorem preceding this conjecture proves the assertion for smooth projective families whose fibres are all of general type. The conjecture asks for invariance in the broader setting of smooth projective deformations.