Categorified Hochschild colored complex conjecture for knots

Let $K\subset S^3$ be a framed oriented knot and let $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$ be a complex of $n$-strand Bar–Natan 1-morphisms. Let $C(K,V_n)$ denote the $V_n$-colored complex from the functorial colored knot homology conjecture, and let $P_n$ be the Cooper–Krushkal categorified Jones–Wenzl idempotent. Categorified Hochschild colored complex conjecture. Each complex $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$ determines a colored complex

$$C'(K,X)\in\operatorname{Ch}^-(\kappa\mathrm{-mod}).$$

The assignment $X\mapsto C'(K,X)$ is functorial in $X$ in the sense that it factors through the dg quantum horizontal trace. Furthermore,

$$C'(K,P_n)\simeq C(K,V_n)$$

up to homotopy. This conjecture seeks a categorified replacement for the trace-based colored knot invariant, with functoriality supplied by the dg quantum horizontal trace; the supplied text gives no proof or resolution.