  We perform an $n_f=2+1$ lattice study of a number of channels where past
claims exist in the literature for the existence of strong-interaction-stable
light-heavy tetraquarks. We find no evidence for any such deeply-bound states,
beyond the $J^P=1^+$, $I=0$ $ud\bar{b}\bar{b}$ and $I=1/2$ $ls\bar{b}\bar{b}$
states already identified in earlier lattice studies. We also describe a number
of systematic improvements to our previous lattice studies, including working
with larger $m_\pi L$ to better suppress possible finite volume effects,
employing extended sinks to better control excited-state contamination, and
expanding the number of operators used in the GEVP analyses. Our results also
allow us to rule out several phenomenological models which predict significant
tetraquark binding in channels where no such binding is found.
