Commutativity conjectures for parabolic affine category O over a deformation ring

Let $R$ be the deformation ring, let $\overline R$ be the corresponding deformation ring after replacing $e$ by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$ and $O^{\nu,\Delta}_{\overline\mu,\overline R}$ be the associated deformed parabolic categories. Let $\theta^{\overline\mu}_\mu$ and $\theta^{\mu'}_{\overline\mu'}$ denote the equivalences between the corresponding categories, and let $F_k,E_k$ and $\overline F_k,\overline E_k$ be the relevant functors. The deformation-ring commutativity conjectures. The two displayed diagrams in the source are commutative: the first compares $F_k$ with the composite $\overline F_{k+1}\overline F_k$, and the second compares $E_k$ with the corresponding composite $\overline E_k\overline E_{k+1}$, under the stated equivalences. These conjectures assert that the level-changing equivalences are compatible with both induction-type and restriction-type functors in the deformed parabolic categories; the source gives no evidence that they have been resolved.