Bounded relative-global klt complements for Fano fibrations

Let $f:X\to Z$ be an $\epsilon$-lc Fano fibration of dimension $d$, meaning that $X$ and $Z$ are normal projective varieties, $f_*\mathcal{O}_X=\mathcal{O}_Z$, $X$ is $\epsilon$-lc, and $-K_X$ is ample over $Z$. A bounded relative-global klt complement is an effective $\mathbb{Q}$-divisor $\Lambda$ on $X$ for which $N(K_X+\Lambda)\sim 0/Z$ and $(X,\Lambda)$ is klt.

Bounded relative-global klt complement conjecture. For every positive integer $d$ and positive real number $\epsilon$, there exists a positive integer $N$, depending only on $d$ and $\epsilon$, such that every $\epsilon$-lc Fano fibration $f:X\to Z$ of dimension $d$ admits an effective $\mathbb{Q}$-divisor $\Lambda$ satisfying

$$N(K_X+\Lambda)\sim 0/Z$$

and $(X,\Lambda)$ is klt.

This is a klt version of the bounded relative-global complement problem and is presented as a variant of Shokurov's conjecture on bounded local complements for Fano fibrations. The supplied abstract states that bounded relative-global klt complements are proved when the base has dimension one; the general assertion in the stated form is not resolved by the supplied text.