We report the results of Phase 1b of the ORGAN experiment, a microwave cavity haloscope searching for dark matter axions in the 107.42-111.93 μeV mass range. The search excludes axions with two-photon coupling g_{aγγ}≥4×10^{-12} GeV^{-1} with 95% confidence interval, setting the best upper bound to date and with the required sensitivity to exclude the axionlike particle cogenesis model for dark matter in this range. This result was achieved using a tunable rectangular cavity, which mitigated several practical issues that become apparent when conducting high-mass axion searches, and was the first such axion search to be conducted with such a cavity. It also represents the most sensitive axion haloscope experiment to date in the ∼100 μeV mass region.