A multi-sequence formula for the Möbius function of the permutation poset

Let $\pi=\pi_1\oplus\cdots\oplus\pi_n$ be a decomposable permutation with $t$ sequences of equal components

$$\pi_{s_i+1}=\cdots=\pi_{s_i+\alpha_i}\ne 1$$

of respective lengths $\alpha_i$, for $1\le i\le t$. Let $\lambda=\lambda_1\oplus\cdots\oplus\lambda_m$ be obtained from $\pi$ by reducing the $i$-th sequence to length $\ell_i$, where $0\le \ell_i\le \alpha_i$. Write $\alpha=\alpha_1+\cdots+\alpha_t$ and $\ell=\ell_1+\cdots+\ell_t$. The multi-sequence Möbius formula. The alternating sum over normal embeddings satisfies

$$\sum_{S\in EZ^{\lambda,\pi}}(-1)^{|S|}=(-1)^{\alpha-\ell-1}\prod_{i=1}^t\binom{\alpha_i-1}{\ell_i-1}.$$

This conjecture extends the formula for removing elements from a single sequence of equal components in the decomposable permutation $\pi$. It gives an explicit product formula for the relevant alternating sum, and hence is intended to describe the Möbius function of the interval $[\lambda,\pi]$ when several equal-component sequences are reduced simultaneously. Its resolution is not established by the supplied context.