Boundedness of klt complements on Fano fibrations over surfaces

Let $(X,B)$ be an ε-lc pair of dimension d with a closed point x∈ X. Birkar 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,ε. We prove that Birkar's conjecture implies Shokurov's conjecture on boundedness of klt complements on Fano fibrations and we confirm Birkar's conjecture in dimension 2. As a corollary, we prove the boundedness of klt complements on Fano fibrations over surfaces.