(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with $|[a,M]| \leq |P|/2$, when $M$ is the maximum of $P$.
Recall that an element a of a poset is join-irreducible if there is no subset $X \subseteq P$ with $a\not\in X$ and $a=\bigvee X$.
Call a (finite) bounded poset $P$ lattice-like in case an element $x \in P$ is join-irreducible iff $x$ is covers a unique element.
Every lattice is lattice-like but not every bounded poset is lattice-like.
Question 1: Is the above conjecture also true for lattice-like posets?
This is true for all such posets with at most 8 points. I would think there is a counterexample but I have not found one yet.
Question 2: Are there attempts in the literature already to generalise the Union-closed sets conjecture from lattices to a larger class of posets?