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An abstract polytope is a poset $X$ (here finite), whose elements are called faces, satisfying these 4 conditions:

1. There is a least face and a greatest face.
2. All flags (i.e. maximal chains) have the same number of faces.
3. $X$ is strongly connected, i.e. for every interval $[F_1,F_2]$ in $X$ and $F, F' \in (F_1,F_2)$ there is a way to get from $F$ to $F'$ via adjacent faces in $(F_1,F_2)$.
4. If $F_1 < F_2$ differ in rank by 2, then there are exactly two intermediate faces $F, F'$ in the interval $(F_1,F_2)$.

Denote the set of faces of rank $i$ in $X$ by $X_i$ and the rank of $X$ by $n$. My question is:

If $X$ is an abstract polytope with $\chi(X) = 0$, i.e. $\sum_{i=-1}^n (-1)^i X_i = 0$, then is $X$ necessarily Eulerian, i.e. is it the case that for every $F_1 < F_2$, $\sum_{i = rank(F_1)}^{rank(F_2)} (-1)^i[F_1,F_2]_i = 0$?

An abstract polytope is a poset $X$ (here finite), whose elements are called faces, satisfying these 4 conditions:

1. There is a least face and a greatest face.
2. All flags (i.e. maximal chains) have the same number of faces.
3. $X$ is strongly connected, i.e. for every interval $[F_1,F_2]$ in $X$ and $F, F' \in (F_1,F_2)$ there is a way to get from $F$ to $F'$ via adjacent faces in $(F_1,F_2)$.
4. If $F_1 < F_2$ differ in rank by 2, then there are exactly two intermediate faces $F, F'$ in the interval $(F_1,F_2)$.

Denote the set of faces of rank $i$ in $X$ by $X_i$ and the rank of $X$ by $n$. My question is:

If $X$ is an abstract polytope with $\chi(X) = 0$, i.e. $\sum_{i=-1}^n (-1)^i X_i = 0$, then is $X$ necessarily Eulerian, i.e. is it the case that for every $F_1 < F_2$, $\sum_{i = rank(F_1)}^{rank(F_2)} (-1)^i[F_1,F_2]_i = 0$?

(Edit: the $X$'s I have in mind are lattices, i.e. any two distinct faces have a unique join [least upper bound] and meet [greatest lower bound].)