  The spin-exotic hybrid meson $\pi_{1}(1600)$ is predicted to have a large
decay rate to the $\omega\pi\pi$ final state. Using 76.6~pb$^{-1}$ of data
collected with the GlueX detector, we measure the cross sections for the
reactions $\gamma p \to \omega \pi^+ \pi^- p$, $\gamma p \to \omega \pi^0 \pi^0
p$, and $\gamma p\to\omega\pi^-\pi^0\Delta^{++}$ in the range $E_\gamma =$ 8-10
GeV. Using isospin conservation, we set the first upper limits on the
photoproduction cross sections of the $\pi^{0}_{1}(1600)$ and
$\pi^{-}_{1}(1600)$. We combine these limits with lattice calculations of decay
widths and find that photoproduction of $\eta'\pi$ is the most sensitive
two-body system to search for the $\pi_1(1600)$.
