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

Question: Is there a non-boolean finite lattice $L$ with the following property? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$

Remark: It follows that $\mu(\hat{0},\hat{1}) = \pm1$.