Categorified generalized projector decomposition conjecture

Let $\kappa$ be the coefficient field, let $\operatorname{Ch}^-(\mathsf{BN}_n)$ be the homotopy category of bounded-above complexes of $n$-strand Bar–Natan 1-morphisms, and let $P_\varepsilon$ be the categorified Temperley–Lieb algebra idempotent constructed by the second author and B. Cooper. For $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$, write $[X]$ for its class in the quantum horizontal trace. Categorified generalized projector decomposition conjecture. For a given $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$, there exist complexes $\hTr_{q,\varepsilon}(X)\in\operatorname{Ch}^-(\kappa\mathrm{-mod})$ such that

$$[X]\simeq\bigoplus_\varepsilon\hTr_{q,\varepsilon}(X)\otimes_\kappa[P_{|\varepsilon|}].$$

Here $[P_\varepsilon]$ is the class of $P_\varepsilon$ in the quantum horizontal trace, and $\hTr_{q,\varepsilon}(X)$ is characterized by

$$[X\star P_\varepsilon]=\hTr_{q,\varepsilon}(X)\otimes[P_\varepsilon],$$

where $\star$ denotes composition of 1-morphisms in $\mathsf{BN}$. This conjecture is the categorified analogue of decomposing a Temperley–Lieb endomorphism into contributions from primitive idempotents grouped by their number of through strands; its status is not resolved in the supplied text.