To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its closed intervals have the same number of odd and even rank elements.
My question is this:
Is there an Eulerian poset $P$ such that the order complex $\Delta(P)$ has a non positive $h$-vector?
I presume that the answer is yes, but I expect examples to be complicated to construct. I would be surprised if nobody thought of this question before, so probably some adequate reference already provides a full answer.
In Stanley's EC1 (2nd ed.), page 459, one can find an example of an Eulerian poset of rank 7, on 82 elements such that the flag $h$-vector of $P$ has a negative entry, but this poset does not provide an answer to my question.
As follows from a comment by Stanley in the same page, all Eulerian posets of rank $\leq 6$ will have a non-negative flag $h$-vector, and hence $h(\Delta(P))$ will be non-negative. In particular, any examples answering negatively my question would be on rank $7$ or more. Let me also say that guesses within the realm of polytopes, CW spheres, Bruhat intervals, etc. will not work, as these are Cohen-Macaulay (and hence the full flag $h$-vector is non-negative).
