Orbit length conjecture for iterated bridge sums of trees and complete graphs

Let $G_1=(V_1,E_1),\ldots,G_q=(V_q,E_q)$ be graphs on $n_1,\ldots,n_q$ vertices, respectively, where each $G_i$ is either a complete graph or a tree. Form $G$ by bridge summing $G_1$ with $G_2$, $G_2$ with $G_3$, and so on, at arbitrary vertices. Let $V'=V_1\cup\cdots\cup V_q$ and let $N=|V'|$. Iterated bridge-sum orbit-length conjecture. For all such $G$, the orbit length of toric promotion on $G$ is $N(N-1)$. This is known when all the summands are trees, but remains open for arbitrary mixtures of trees and complete graphs.