Bounded multiplicities for irreducible fibers of Fano fibrations

Let $d$ be a positive integer and $\epsilon$ a positive real number. Let $X$ be a normal variety of dimension $d$ and let $\pi:X\to C$ be a contraction to a curve $C$. Assume that $X$ is $\epsilon$-lc, $-K_X$ is ample over $C$, and every fiber of $X\to C$ is irreducible. For a closed point $z\in C$, write

$$\pi^*z=m_zF_z,$$

where $F_z$ is the irreducible component of $\pi^*z$. Irreducible-fiber multiplicity conjecture. There exists a positive integer $m$, depending only on $d$ and $\epsilon$, such that $m_z\le m$. This is the Fano-fibration multiplicity conjecture used in the paper's reductions; no general proof is given in the supplied text.