  We study positive braid knots (the knots in the three-sphere realized as
positive braid closures) through the lens of the L-space conjecture. This
conjecture predicts that if $K$ is a non-trivial positive braid knot, then for
all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along
$K$ admits a taut foliation. Our main result provides some positive evidence
towards this conjecture: we construct taut foliations in such manifolds
whenever $r<g(K)+1$. As an application, we produce a novel braid positivity
obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a knot $K$
is braid positive if and only if $K$ is the unknot. We also present some
curious examples demonstrating the limitations of our construction; these
examples can also be viewed as providing some negative evidence towards the
L-space conjecture. Finally, we apply our main result to produce taut
foliations in some splicings of knot exteriors.
