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.
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