  Universal solutions to deformation quantization problems can be conveniently
classified by the cohomology of suitable graph complexes. In particular, the
deformation quantizations of (finite-dimensional) Poisson manifolds and Lie
bialgebras are characterised by an action of the Grothendieck-Teichm\"uller
group via one-colored directed and oriented graphs, respectively. In this note,
we study the action of multi-oriented graph complexes on Lie bialgebroids and
their "quasi" generalisations. Using results due to T. Willwacher and M.
Zivkovi\'c on the cohomology of (multi)-oriented graphs, we show that the
action of the Grothendieck-Teichm\"uller group on Lie bialgebras and quasi-Lie
bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two
colors, one of them being oriented. However, this action generically fails to
preserve the subspace of Lie bialgebroids. By resorting to graphs with two
oriented colors, we instead show the existence of an obstruction to the
quantization of a generic Lie bialgebroid in the guise of a new
$\mathsf{Lie}_\infty$-algebra structure non-trivially deforming the "big
bracket" for Lie bialgebroids. This exotic $\mathsf{Lie}_\infty$-structure can
be interpreted as the equivalent in $d=3$ of the Kontsevich-Shoikhet
obstruction to the quantization of infinite-dimensional Poisson manifolds (in
$d=2$). We discuss the implications of these results with respect to a
conjecture due to P. Xu regarding the existence of a quantization map for Lie
bialgebroids.
