The outerplane adjacency-polytope volume formula

Let $G$ be an outerplane graph. For each connected component $G_i$ of $G$, let $B_{i,1},\dots,B_{i,b_i}$ be its blocks; let $\mathscr{F}(B_{i,j})$ denote the bounded faces of $B_{i,j}$, let $\operatorname{ewd}{B_{i,j}}$ denote its extended weak dual, and let $v_F$ be the vertex of this dual corresponding to $F$. The normalized volume of the type-PQ adjacency polytope is denoted by $\operatorname{NVol}(\nabla^{\operatorname{PQ}}_G)$. Outerplane adjacency-polytope volume conjecture. For any outerplane graph $G$,

$$\operatorname{NVol}(\nabla^{\operatorname{PQ}}_G) = \prod_{i=1}^k \prod_{j=1}^{b_i} 2^{|V(B_{i,j})| - 2|\mathscr{F}(B_{i,j})|}\prod_{F \in \mathscr{F}(B_{i,j})} \deg_{\operatorname{ewd}{B_{i,j}}}(v_F).$$

The formula is proved for a proper but large class of outerplane graphs, namely those whose bounded faces are adjacent to the outer face, while experimental data suggests that it holds for all outerplane graphs; a proof remains open.