Leveraging the features of the GstLAL pipeline, we present the results of a matched filtering search for asymmetric binary black hole systems with heavily misaligned spins in LIGO and Virgo data taken during the third observing run. Our target systems show strong imprints of precession whereas current searches have nonoptimal sensitivity in detecting them. After measuring the sensitivity improvement brought by our search over standard spin-aligned searches, we report the detection of 30 gravitational wave events already discovered in the latest version of the Gravitational Wave Transient Catalog. However, we do not find any additional significant gravitational wave candidates. Our results allow us to place an upper limit of R_{90%}=0.28_{-0.04}^{+0.33} Gpc^{-3} yr^{-1} on the merger rate of a hypothetical subpopulation of asymmetric, heavily precessing signals, not identified by other searches. Since our upper limit is consistent with the latest rate estimates from the LIGO-Virgo-KAGRA Collaboration, our findings rule out the existence of a yet-to-be-discovered population of precessing binaries.