Birkar–Shokurov's conjecture on lc divisors over Fano fibrations

Let $d$ be a natural number and let $\epsilon$ be a positive real number. Assume $(X,B)$ is an $\epsilon$-lc pair of dimension $d$, $f:X\to Z$ is a contraction with $\dim Z>0$, $z\in Z$ is a closed point, $X$ is of Fano type over $Z$, and $-(K_X+B)$ is nef over $Z$. Birkar–Shokurov's conjecture. There is a positive real number $t$ depending only on $d,\epsilon$ such that there exists an effective Cartier divisor $H$ on a neighbourhood $U$ of $z$ in $Z$, with $z\in\operatorname{Supp}H$, for which $(X,B+tf^*H)$ is lc over $U$. This conjecture is one of two equivalent conjectures proposed to establish boundedness of klt complements; the source records several special cases but leaves the general statement open.