Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

Question: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

It is obviously satisfied by the boolean lattices. Now a boolean lattice is the face lattice of a simplex and John's comment below suggests that it could be satisfied by the face lattice of an arbitrary (convex?) polytope. Is it true? Can we extend to any Eulerian lattice? Anything else?