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 $0<1$)?



Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$
No. The counterexample to [this](https://mathoverflow.net/questions/211919/simplyfing-join-incomplete-lattice-endomorphisms/211922#211922) question also works here. 


