Let $P$ be a finite connected poset with incidence algebra $A_P$. For the definition and results on Koszul algebras for incidence algebras, see for example here

Question: Which posets have the property that $A_P$ is a Koszul algebra such that the Koszul dual of $A_P$ is isomorphic to $A_P$?

It seems this property is extremely rare and only the finest posets are in this class such as Boolean lattices (and no other distributive lattices ?!) or the strong Bruhat order for the symmetric group.