The fan structure conjecture for diagrams and strong Khovanskii bases

Let $\Sigma$ be a fan, let $I$ be the ideal defining the fiber data, and let $\Delta(\Sigma, \operatorname{Trop}_\star(I))$ denote the set of diagrams for bundles over the toric variety $Y(\Sigma)$ with fiber $E$ and Khovanskii basis $\mathcal{B}\subset \operatorname{Sym}(E)$. Fan structure conjecture. The set $\Delta(\Sigma, \operatorname{Trop}_\star(I))$ is the support of a fan $\mathcal{F}$ such that the set of diagrams corresponding to bundles for which $\mathcal{B}$ is a strong Khovanskii basis is a union of faces of $\mathcal{F}$. This describes the strong Khovanskii-basis condition polyhedrally and supports the study of finite generation and Mori dream space properties for projectivized toric vector bundles.