Large filters of quasiorder lattices can be generated by few elements

For a poset $(P;\le )$, the quasiorders (AKA preorders) extending the poset order “$\le$” form a complete lattice $F$, which is a filter in the lattice of all quasiorders of the set $P$. We prove that if the poset order “$\le$” is small, then $F$ can be generated by few elements.