  We study the exclusive photoproduction of a $ \pi ^{0}\gamma $ pair with
large invariant mass $ M_{\gamma \pi}^2 $, which is sensitive to the exchange
of either two quarks or two gluons in the $ t $-channel. In this letter, we
show that the process involving two-gluon exchanges does not factorize in the
Bjorken limit at the leading twist. This can be explicitly demonstrated by the
fact that there exist diagrams, which contribute at the leading twist, for
which Glauber gluons remain trapped, due to the pinching of the contour
integrations of both the plus and minus component of the Glauber gluon
momentum. For the same reason, $\pi^0$-nucleon scattering to two photons also
suffers from the same issue. On the other hand, we stress that there are no
issues with respect to collinear factorization for the quark channels. By
considering an analysis of all potential reduced diagrams of leading pinch
surfaces, we argue that the quark channel is safe from Glauber pinches, and
therefore, a collinear factorization in that case follows through without any
problems. This means that processes where gluon exchanges are forbidden, such
as the exclusive photoproduction of $ \pi ^{\pm}\gamma $ and $ \rho^{0,\,\pm}
\gamma $, are unaffected by the factorization breaking effects we point out in
this letter.
