Siu's conjecture on deformation invariance of plurigenera for compact Kähler manifolds

Let $\pi:X\to\Delta$ be a smooth family of compact Kähler manifolds over the unit disk in $\mathbb{C}$, with fiber $X_t$. For a positive integer $m\geq 1$, define the $m$-genus by

$$P_m(X_t):=\operatorname{dim}H^0(X_t,mK_{X_t}).$$

Siu's conjecture. The $m$-genus $P_m(X_t)$ is independent of $t\in\Delta$. The problem is widely open for smooth Kähler families, although deformation invariance is known in several projective and Moishezon settings and recent work gives partial progress.