The projector relation for braided trace powers

Let ${\rm Tr}_R L^k\in {\cal L}^k(R)$ be the braided trace power, and let $P_+^{(k)}$ and $P_{+2}^{(k-1)}$ be the symmetrizer operators acting on the corresponding tensor powers. Projector relation. For every positive integer $k$ for which these operators are defined, the relation

$$P_+^{(k)}{\rm Tr}_R L^k=P_{+2}^{(k-1)}{\rm Tr}_R L^k$$

holds. This identity is used to put the homogeneous element ${\rm Tr}_R L^k$ into the canonical form required before applying the differential in the complex of braided differential forms. The source presents the relation without resolving its general status; the preceding discussion indicates that explicit projector constructions are known only in low degrees.