Positive braid closures and taut foliations

We study taut foliations on the complements of non-split positive braid closures in $S^3$. If $L$ is such a link with components $L_1,\dots ,L_n$ and at least one component is not the unknot, then the Dehn surgery along a multislope $(s_1,\dots ,s_n)\in {\mathbb{Q}}^n$ satisfying $s_i<2g(L_i)-1$ for $i=1,2,\dots ,n$ yields a non-L-space that admits a co-oriented taut foliation.