Möbius-value conjecture for staircase partition posets

Let $\lambda$ be a partition with repeating elements, and let ${\mathbb D}_{\mathsf{S}}(\lambda)$ be the associated poset of staircase matrices. Denote its Möbius function by $\mu^{{\mathbb D}_{\mathsf{S}}(\lambda)}(x,y)$ for comparable elements $x$ and $y$. Möbius-value conjecture. For all comparable elements $x$ and $y$ in ${\mathbb D}_{\mathsf{S}}(\lambda)$, one has $\mu^{{\mathbb D}_{\mathsf{S}}(\lambda)}(x,y)\in\{-1,0,1\}$. The conjecture has been verified in several specific cases and is important for determining coefficients in the Cauchy identity for staircase matrices; it follows from the shellability conjecture above.