Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$

Question

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$

Is it true that if there is a join-incomplete lattice homomorphism $f:L\to L$, then there is also a join-incomplete lattice homomorphism $f_2:L\to {\bf 2}$ (where ${\bf 2} = \{0,1\}$ oredered by $0<1$)?

Answer 1

No. The counterexample to this question also works here.