Grothendieck's standard conjecture of Künneth type

Let $k$ be a base field, let $H^*$ be a fixed classical Weil cohomology with realization functor $R$, and let $X$ be a variety over $k$ of dimension $d$. Write $\mathfrak{h}(X)$ for its homological motive. A decomposition is required to realize the cohomological grading, and $\vee$ denotes the dual motive while $(-d)$ denotes a Tate twist.

Standard conjecture of Künneth type. There exists a unique decomposition

$$\mathfrak{h}(X)=\bigoplus_{n=0}^{2d}\mathfrak{h}^n(X)$$

such that

$$R(\mathfrak{h}^n(X))=H^n(X),\qquad \mathfrak{h}^n(X)=\mathfrak{h}^{2d-n}(X)^\vee(-d).$$

This is one of Grothendieck's standard conjectures: it would provide algebraic projectors realizing the Künneth components of the cohomology and a motivic form of Poincaré duality. The source recalls it as a classical conjecture, but gives no resolution here.