Bounded fiber multiplicities for Fano fibrations over curves

Let $d$ be a positive integer and $\epsilon$ a positive real number. Let $(X,B)$ be a log pair of dimension $d$ and let $\pi:X\to C$ be a contraction, where $X$ is of Fano type over the curve $C$ and $K_X+B\sim_{\mathbb{R},C}0$. Assume that $(X,B)$ is $\epsilon$-lc. For a closed point $z\in C$, write

$$\pi^*z=\sum_i m_iF_i,$$

where the $F_i$ are the irreducible components of $\pi^*z$. Bounded fiber multiplicity conjecture. There exists a positive integer $m$, depending only on $d$ and $\epsilon$, such that $m_i\le m$ for every $i$ and every closed point $z\in C$. This conjecture is presented as a direct corollary of the McKernan–Shokurov conjecture and is not resolved in general.