Siu's invariance of plurigenera conjecture for Kähler families

Let $\pi:\mathcal{X}\longrightarrow\Delta$ be a holomorphic family of compact Kähler manifolds $X_t:=\pi^{-1}(t)$ over the open unit disc $\Delta\subset\mathbb{C}$. For a positive integer $m$, write

$$p_m:=\dim_{\mathbb{C}} H^0(X_t,mK_{X_t})$$

for the $m$-th plurigenus of $X_t$. Siu's conjecture. For every positive integer $m$, the plurigenus $p_m$ is independent of $t\in\Delta$.

This conjecture proposes invariance of all plurigenera in holomorphic families with compact Kähler fibres, extending the known projective case. The source attributes it to Siu; its resolution status is not established by the supplied text.