Several unforeseen collective phenomena have been observed in high-multiplicity small collision systems that resemble the well-established signatures of the quark-gluon plasma (QGP) formation in heavy-ion collisions. However, jet quenching effects have not been observed in small collision systems. Quantification or setting limits on the magnitude of jet quenching in small systems is essential for understanding the limits of the QGP formation. These proceedings present the outcomes of a search for jet quenching effects performed by the ALICE collaboration in pp collisions at $\sqrt{s} = 13$ TeV as a function of charged-particle multiplicity, measured in forward rapidity. Two jet observables are studied: inclusive $p_{\text{T}}$-differential jet cross section, and the semi-inclusive yield of jets recoiling from a high-$p_{\text{T}}$ trigger-hadron. Jets are reconstructed from charged-particle tracks using the anti-$k_{\text{T}}$ algorithm with resolution parameter $R$ in the range $0.2 - 0.6$. To search for jet quenching effects, both analyses compare jet yields measured in different multiplicity intervals. The analysis of inclusive jets reveals that the rise of event activity leads to an increase in jet production with a weak impact on the spectra slope for high-$p_{\text{T}}$ jets. In the semi-inclusive analysis, the acoplanarity distribution of recoil jets measured in high-multiplicity events exhibits a substantial suppression and broadening when compared to the corresponding spectrum obtained from minimum-bias events. These peculiar features are also seen in pp events simulated by the PYTHIA 8 Monte Carlo event generator. Further studies of the PYTHIA 8 data suggest that the observed suppression and broadening arise from a bias posed by the ALICE high-multiplicity trigger. This bias leads to a growth of the probability to measure high-$p_{\text{T}}$ recoil jets in the acceptance of the forward V0 detector. Furthermore, the high-multiplicity trigger biases toward final states with multi-jet topology.