  Suppose $(B,\pi)$ is an open book supporting $(Y,\xi)$, where the binding $B$
is possibly disconnected, and $K$ is a braid about this open book. Then $B\cup
K$ is naturally a transverse link in $(Y,\xi)$. We prove that the transverse
link invariant in knot Floer homology, \[\widehat{t}(B\cup K)\in
\widehat{HFK}(-Y,B\cup K),\] defined in [BVV13] is always nonzero. This
generalizes the main results of Etnyre and Vela-Vick in [VV11, EVV10]. As an
application, we show that if $K$ is braided about an open book with connected
binding, and has fractional Dehn twist coefficient greater than one, then
$\widehat{t}(K)\ne 0$. This generalizes a result of Plamenevskaya [PLA15] for
classical braids.
