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 $f(\bigvee_L S) \neq \bigvee_L f(S).$

If $f:L\to L$ is join-incomplete, does there exist a lattice homomorphism $f:L\to L$ such that

• there is $S\subseteq L$ with $f(S) \subseteq S$, and
• $f(\bigvee_L S) > \bigvee_L f(S)$

?