On a connectedness principle of Shokurov-Koll\'ar type

Let $(X,\mathrm{\Delta })$ be a log pair over $S$, such that $-(K_X+\mathrm{\Delta })$ is nef over $S$. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of $(X,\mathrm{\Delta })$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension $\le 4$ and in arbitrary dimension assuming the termination of klt flips.