The strong Nakano property in Banach lattices

We study the strong Nakano property in Banach lattices with a special focus on free Banach lattices. We show that for every finite dimensional Banach space $E$, the free Banach lattice $FBL[E]$ has the strong Nakano property with a constant independent of the dimension. It is also shown that if $FBL[E]$ has the strong Nakano property, then $E$ cannot contain subspaces isomorphic to neither $c_0$ nor $L_1$.