Evidence for the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3872</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:mi>ω</mml:mi></mml:math>

We present a study of the decays ${B}^{0,+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{K}^{0,+}$, using $467\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs recorded with the BABAR detector. We present evidence for the decay mode $X(3872)\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\omega}$, with product branching fractions $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}X(3872){K}^{+})\ifmmode\times\else\texttimes\fi{}\mathcal{B}(X(3872)\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\omega})=[0.6\ifmmode\pm\else\textpm\fi{}0.2(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.1(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, and $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}X(3872){K}^{0})\ifmmode\times\else\texttimes\fi{}\mathcal{B}(X(3872)\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\omega})=[0.6\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.1(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. A detailed study of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ mass distribution from $X(3872)$ decay favors a negative-parity assignment.