We use QCD Laplace sum-rules to predict masses of open-flavour heavy-lighthybrids where one of the hybrid's constituent quarks is a charm or bottom andthe other is an up, down, or strange. We compute leading-order, diagonalcorrelation functions of several hybrid interpolating currents, taking intoaccount QCD condensates up to dimension-six, and extract hybrid masspredictions for all $J^P\in\{0^{\pm},\,1^{\pm}\}$, as well as explore possiblemixing effects with conventional quark-antiquark mesons. Within theoreticaluncertainties, our results are consistent with a degeneracy between theheavy-nonstrange and heavy-strange hybrids in all $J^P$ channels. We find asimilar mass hierarchy of $1^+$, $1^{-}$, and $0^+$ states (a $1^{+}$ statelighter than essentially degenerate $1^{-}$ and $0^{+}$ states) in both thecharm and bottom sectors, and discuss an interpretation for the $0^-$ states.If conventional meson mixing is present the effect is an increase in the hybridmass prediction, and we estimate an upper bound on this effect.