  The D0 Collaboration has recently reported a narrow peak structure in the
$B_s^0\pi^\pm$ invariant-mass spectrum, that is called $X(5568)$. Its proposed
quantum numbers $J^P=0^+/1^+$ are not determined unambiguously, and we show
that one can fit the data assuming $J^P=1^-$ with the same mass and width for
the resonance signal. The corresponding isovector $P$-wave coupled-channel
scattering involving the states $B_s\pi$, $B_s^*\pi$, $B \overline{K}$ and
$B^*\overline K$ is studied employing leading-order Heavy Meson Chiral
Perturbation Theory, which is then unitarized making use of standard techniques
from Unitary Chiral Perturbation Theory. The subtraction constants that appear
in the unitarization process are determined by a $u$-channel crossing symmetry
constraint or by naturalness arguments, being the numerical values obtained in
both cases quite closed, and no further freedom is left in the model. A crucial
point is that if the $X(5568)$ is $1^-$ and decaying into $B_s^0\pi^\pm$ then
it should appear as a pole in the $T$-matrix elements. It finally turns out
that no $X(5568)$ pole is found that can be qualified as dynamically generated
from the isovector $P$-wave coupled-channel dynamics. This result disfavors the
interpretation of the $X(5568)$ within a "molecular" picture.
