Functorial colored knot homology conjecture

Let $K$ be a framed oriented knot, let $\operatorname{Kar}(\mathcal T\mkern-3mu\mathcal L)^\oplus$ be the additive Karoubi envelope of the Temperley–Lieb category, and let $V_n$ denote the $n$-th color with $V_1$ the fundamental color. Functorial colored knot homology conjecture. Each framed oriented knot $K$ determines a dg functor

$$C(K;-):\operatorname{Ch}^-(\operatorname{Kar}(\mathcal T\mkern-3mu\mathcal L)^\oplus)\longrightarrow\operatorname{Ch}^-(\kappa\mathrm{-mod})$$

such that $C(K;V_1^{\otimes n})$ is the quantum annular complex associated to the $n$-cable of $K$, and $C(K;V_n)$ is a quantum annular version of Khovanov's colored $\mathfrak{sl}_2$-homology. This is proposed as a functorial categorified Reshetikhin–Turaev construction; the supplied text says it is planned for future work and does not state that it has been proved.