The Shokurov–Kollár connectedness conjecture for non-klt loci

Let $k$ be an algebraically closed field of characteristic zero. Let $\pi:X\to S$ be a proper morphism with connected fibers, and let $(X,\Delta)$ be a log pair. The non-klt locus $\operatorname{Nklt}(X,\Delta)$ is the locus where $(X,\Delta)$ is not Kawamata log terminal. Assume that

$$-(K_X+\Delta)\text{ is }\pi\text{-nef}.$$

Shokurov–Kollár connectedness conjecture. For every point $s\in S$, the fiber of the non-klt locus has at most two connected components:

$$\pi^{-1}(s)\cap\operatorname{Nklt}(X,\Delta)$$

has at most two connected components.

The usual connectedness lemma gives connected fibers when $-(K_X+\Delta)$ is also $\pi$-big; this conjecture predicts the optimal replacement when only relative nefness is assumed. It is proved in dimension at most four and in arbitrary dimension assuming termination of klt flips.