  We present the results of the largest $L^{\prime}$ ($3.8~\mu$m) direct
imaging survey for exoplanets to date, the Large Binocular Telescope
Interferometer (LBTI) Exozodi Exoplanet Common Hunt (LEECH). We observed 98
stars with spectral types from B to M. Cool planets emit a larger share of
their flux in $L^{\prime}$ compared to shorter wavelengths, affording LEECH an
advantage in detecting low-mass, old, and cold-start giant planets. We
emphasize proximity over youth in our target selection, probing physical
separations smaller than other direct imaging surveys. For FGK stars, LEECH
outperforms many previous studies, placing tighter constraints on the hot-start
planet occurrence frequency interior to $\sim20$ au. For less luminous,
cold-start planets, LEECH provides the best constraints on giant-planet
frequency interior to $\sim20$ au around FGK stars. Direct imaging survey
results depend sensitively on both the choice of evolutionary model (e.g., hot-
or cold-start) and assumptions (explicit or implicit) about the shape of the
underlying planet distribution, in particular its radial extent. Artificially
low limits on the planet occurrence frequency can be derived when the shape of
the planet distribution is assumed to extend to very large separations, well
beyond typical protoplanetary dust-disk radii ($\lesssim50$ au), and when
hot-start models are used exclusively. We place a conservative upper limit on
the planet occurrence frequency using cold-start models and planetary
population distributions that do not extend beyond typical protoplanetary
dust-disk radii. We find that $\lesssim90\%$ of FGK systems can host a 7 to 10
$M_{\mathrm{Jup}}$ planet from 5 to 50 au. This limit leaves open the
possibility that planets in this range are common.
