Chow–Künneth projector conjecture for the $(2n-2)$-nd motive

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$. Let $M_2(X)^*$ and $i_{2n-2}:M_2(X)^*\otimes {\bf L}^{n-2}\to M(X)$ be as obtained from the decomposition of $\underline{\operatorname{Hom}}({\bf L}^{n-2},M(X))$. Chow–Künneth projector conjecture. There is a morphism $p_{2n-2}:M(X)\to M_2(X)^*\otimes {\bf L}^{n-2}$ in ${\rm DM}_{\bf Q}^{e\!f\!f}$ such that $p_{2n-2}i_{2n-2}={\rm id}$, $i_{2n-2}p_{2n-2}$ induces the Künneth projector onto $H^{2n-2}(X,{\bf Q})$, and its dual induces the Künneth projector onto $H^2(X,{\bf Q})$. This is a conjectural motivic realization of the relevant Künneth projectors; the source gives no resolution status.