  Spectroscopic parameters and widths of the fully open-flavor axial-vector and
pseudoscalar tetraquarks $X_{\mathrm{AV}}$ and $X_{\mathrm{PS}}$ with content
$[ud][\overline{c}\overline{s}]$ are calculated by means of the QCD sum rule
methods. Masses and current couplings of $X_{\mathrm{AV}}$ and $X_{
\mathrm{PS}}$ are found using two-point sum rule computations performed by
taking into account various vacuum condensates up to dimension $10$. The full
width of the axial-vector state $X_{\mathrm{AV}}$ is evaluated by including
into analysis $S$-wave decay modes $X_{\mathrm{AV}}\to D^{\ast
}(2010)^{-}K^{+}$, $\overline{D}^{\ast }(2007)^{0}K^{0}$, $D^{-}K^{\ast
}(892)^{+}$, and $\overline{D}^{0}K^{\ast }(892)^{0}$. In the case of $X_{
\mathrm{PS}}$, we consider $S$-wave decay $X_{\mathrm{PS}}\to \overline{D}
_{0}^{\ast }(2300)^{0}K^{0}$, and $P$-wave processes $X_{\mathrm{PS}}\to
D^{-}K^{\ast}(892)^{+}$ and $X_{\mathrm{PS}}\to \overline{D}
^{0}K^{\ast}(892)^{0}$. To determine partial widths of these decay modes, we
employ the QCD light-cone sum rule method and soft-meson approximation, which
are necessary to estimate strong couplings at tetraquark-meson-meson vertices
$X_{\mathrm{AV}}D^{-}D^{\ast }(2010)^{-}K^{+} $, etc. Our predictions for the
mass $m_{\mathrm{AV}}=(2800 \pm 75)~\mathrm{MeV}$ and width
$\Gamma_{\mathrm{AV}}=(58 \pm 10)~\mathrm{MeV}$ of the tetraquark $X_{
\mathrm{AV}}$, as well as results $m_{\mathrm{PS}}=(3000 \pm 60)~\mathrm{MeV} $
and $\Gamma_{\mathrm{PS}}=(65 \pm 12)~\mathrm{MeV}$ for the same parameters of
$X_{\mathrm{PS}}$ may be useful in future experimental studies of multiquark
hadrons.
