  Using proton-proton collisions at centre-of-mass energies $7$, $8$, and $13$
TeV, with a total integrated luminosity of $9\,\text{fb}^{-1}$, the LHCb
collaboration has performed amplitude analyses of the $B^+\to D^+D^-K^+$,
$B^+\to D^- D_s^+ \pi^+$ and $B^0\to \bar{D}^0 D_s^+ \pi^-$ decays, observing
that new $T_{cs}$ and $T_{c\bar s}$ resonances are required in order to explain
the experimental data. These signals could be the first observation of
tetraquark candidates that do not contain a heavy quark-antiquark pair; in
fact, they consist of four different flavours of quarks, one of which is a
doubly charged open-charm state. We present herein an analysis of the $T_{cs}$
and $T_{c\bar s}$ states, which is an extension of our recently published study
of similar $T_{cc}^+$ exotic candidates. Our theoretical framework is a
constituent-quark-model-based coupled-channels calculation of $qq^\prime \bar s
\bar c$ and $cq\bar s\bar q^{\prime}$ tetraquark sectors for $T_{cs}$ and
$T_{c\bar s}$ structures, respectively. We explore the nature, and pole
position, of the singularities that appear in the scattering matrix with
spin-parity quantum numbers: $J^P=0^\pm$, $1^\mp$, and $2^\pm$. The constituent
quark model has been widely used in the heavy quark sector, and thus all model
parameters are already constrained from previous works. This makes our
predictions robust and parameter-free. We find many singularities in the
solution of various scattering-matrix problems which are either virtual states
or resonances, but not bound states. Some of them fit well with the
experimental observations of the spin-parity, mass and width of $T_{cs}$ and
$T_{c\bar s}$ candidates, and thus tentative assignments are made; however,
with caution, because the experimental Breit-Wigner parameters are related to
the pole characteristics.
