Peckness conjecture for edge posets of Boolean algebra quotients

Let $B_n$ be the poset of subsets of $\{1,\ldots,n\}$ ordered by containment, and let $G\subseteq\operatorname{Aut}(B_n)$ act order-preservingly and rank-preservingly on $B_n$. The edge poset $\mathcal E(P)$ of a graded poset $P$ is the graded poset whose elements are the edges in the Hasse diagram of $P$.

Peckness conjecture. The edge poset of the quotient satisfies

$$\mathcal E(B_n/G)\text{ is Peck}.$$

Since $B_n$ is unitary Peck and quotients by such group actions are Peck, this conjecture asks whether the corresponding edge posets retain Peckness after passing to Boolean algebra quotients. The source gives no resolution, so the conjecture remains open.