Summary: The congruence properties close to lower boundedness in the sense of McKenzie are treated. In particular, an affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety \({\mathcal K}\), the lattice \(L_q({\mathcal K})\) possesses the Day-Pudlak-T\D{u}ma property. But if a quasivariety is locally finite (without the condition of being finitely generated), that lattice satisfies only the Pudlak-T\D{u}ma property.