In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements.

Briefly, this follows by arranging the subposet $J(L) = \{x_1,\dots,x_n\} \subset L$ so that $j > i \implies x_j \nleq x_i$, and then observing that $0 < x_1 < x_1 \lor x_2 < \dots < x_1 \lor \dots \lor x_n=1$ because join-irreducible elements are join-prime in distributive lattices.

What other natural classes of finite lattices $L$ satisfy $|J(L)| \leq height(L)$?

For example, do join-semidistributive lattices have this property?

Many thanks.