Naturality and decomposability conjectures for the categorified ribbon element

Let $\dot{\mathcal{U}}$ be the Karoubi envelope of the categorified quantum group $\mathcal{U}$, and let $r\mathbf{1}_n$ and $r^{-1}\mathbf{1}_n$ be the bicomplexes categorifying the ribbon element and its inverse. For each complex $X\in\operatorname{Com}(\mathcal{U})$, let $\kappa_X:Xr\mathbf{1}_n\rightarrow rX\mathbf{1}_n$ and $\eta_X:Xr^{-1}\mathbf{1}_n\rightarrow r^{-1}X\mathbf{1}_n$ be the chain maps from the centrality theorem. A bicomplex is indecomposable if it is not isomorphic to a nontrivial direct sum, and a complex is contractible if its identity map is chain-homotopic to zero.

Naturality and decomposability conjectures. The chain maps $\kappa_X$ and $\eta_X$ are natural, namely, they commute with all 2-morphisms of $\dot{\mathcal{U}}$. Moreover, for $n\geq 0$, the bicomplex $r\mathbf{1}_n$ is indecomposable in $\operatorname{Kom}(\dot{\mathcal{U}})$, and $\omega(r\mathbf{1}_{-n})$ is isomorphic to a direct sum of $r\mathbf{1}_n$ and a contractible complex. For $n\leq 0$, $\omega(r\mathbf{1}_{-n})$ is indecomposable in $\operatorname{Kom}(\dot{\mathcal{U}})$, and $r\mathbf{1}_n$ is isomorphic to a direct sum of $\omega(r\mathbf{1}_{-n})$ and a contractible complex. For $n=0$, the bicomplexes $r\mathbf{1}_n$ and $\omega(r\mathbf{1}_{-n})$ are isomorphic.

These expected properties would strengthen the categorified ribbon element's centrality and clarify its indecomposable decomposition, but the source provides no resolution of either assertion.