  Search strategies for the third-generation leptoquarks (LQs) are distinct
from other LQ searches, especially when they decay to a top quark and a $\tau$
lepton. We investigate the cases of all TeV-scale scalar and vector LQs that
decay to either a top-tau pair (charge-$1/3$ and $5/3$ LQs) or a top-neutrino
pair (charge-$2/3$ LQs). One can then use the boosted top (which can be tagged
efficiently using jet-substructure techniques) and high-$p_{\rm T}$ $\tau$
leptons to search for these LQs. We consider two search channels with either
one or two taus along with at least one hadronically decaying boosted top
quark. We estimate the high luminosity LHC (HL-LHC) search prospects of these
LQs by considering both symmetric and asymmetric pair and single production
processes. Our selection criteria are optimised to retain events from both pair
and single production processes. The combined signal has better prospects than
the traditional searches. We include new three-body single production processes
to enhance the single production contributions to the combined signal. We
identify the interference effect that appears in the dominant single production
channel of charge-$1/3$ scalar LQ ($S^{1/3}$). This interference is
constructive if $S^{1/3}$ is weak-triplet and destructive, for a singlet one.
As a result, their LHC prospects differ appreciably.
