The Möbius function formula conjecture for intervals $[M_m,\pi]$

Let $M_m=24\ldots(2m)13\ldots(2m-1)$, and let $\pi$ be a permutation in $\mathcal{P}_0^n\cup\mathcal{P}_1^n$. Define the statistics $\lambda$, $\sigma$, $j^a$, $j^b$, $\epsilon$, $s$, $s_\theta$, and $\widehat{C}$ as in the preceding construction. In particular,

$$\widehat{C}_\beta^\alpha(k,s)=\begin{cases}\binom{\alpha-2k}{\beta},&0\leqslant k<\frac{s}{2},\\binom{\alpha-2(s-k)+1}{\beta},&\frac{s}{2}\leqslant k<s.\end{cases}$$

Möbius function formula conjecture. If $\pi$ begins with $12$, ends with $(n-1)n$, or contains a triple adjacency, then

$$\mu(M_m,\pi)=0.$$

Otherwise,

$$\begin{aligned} |\mu(M_m,\pi)|={}&\binom{\lambda}{\sigma}-\sum_{\tau=0}^{|j^b|-1}\sum_{\gamma=0}^{\tau}\sum_{\omega=\tau-\gamma}^{j^b_\gamma+s_b(\tau+1,|j^b|-1)-1}\widehat{C}^{\lambda-\tau-2}_{\sigma-\tau-1}(\omega,s)\\ &+\sum_{\tau=0}^{|j^a|-1}\left[\sum_{\gamma=1}^{j^a_\tau+s_a(0,\tau-1)}\widehat{C}^{\lambda-|j^b|-\tau}_{\sigma-|j^b|-\tau}(\gamma,s+1)+\sum_{\omega=1}^{\epsilon}\widehat{C}^{\lambda-|j^b|-\tau}_{\sigma-|j^b|-\tau}(\omega+1,s+1)\right]. \end{aligned}$$

Moreover, the sign of $\mu(M_m,\pi)$ is positive if and only if $n$ is even.

This conjectural formula gives an explicit Möbius value for a broad class of permutation-poset intervals. It was checked computationally for all pairs $(m,n)$ with $m<6$ and $n<12$; the general formula remains unproved in the supplied source.