Integral commutativity conjectures for categorical induction and restriction functors

Let $R$ be the integral coefficient ring, let $\overline R$ be the corresponding coefficient ring for the construction with $e$ replaced by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$ denote the indicated subcategories. Use the equivalences $\theta_\mu^{\overline\mu}$ and $\theta_{\overline\mu'}^{\mu'}$ described in the source, and write $\overline\mu^0$ and $\mu'$ for the intermediate and target weights occurring in the diagrams. The integral commutativity conjectures. The two displayed diagrams involving $\overline F_k,\overline F_{k+1},F_k$ and $\overline E_k,\overline E_{k+1},E_k$, respectively, are commutative. These integral statements refine the preceding comparison to the categories $O^{\nu,\Delta}_{\mu,R}$ and are motivated by the construction of the equivalences over a field and their integral version; the supplied text does not state whether they have been proved.