Boundedness of klt complements on Fano fibrations over surfaces

Let $(X,B)$ be an $\epsilon$-lc pair of dimension $d$ with a closed point $x\in X$. Birkar and Shokurov conjectured that there is an effective Cartier divisor $H$ passing through $x$ such that $(X,B+tH)$ is lc near $x$, where $t$ is a positive real number depending only on $d,\epsilon$. We prove that this conjecture is equivalent to Shokurov’s conjecture on boundedness of klt complements on Fano fibrations and we confirm it in dimension 2. As a corollary, we prove the boundedness of klt complements on Fano fibrations over surfaces.