Bloch–Beilinson co-generation conjecture for birational motives

Let $X$ be a smooth projective variety and let $i$ be a positive integer. Assume that $\mathrm{H}^{*}_{\mathrm{tr}}(X)$ is generated by $\mathrm{H}^i_{\mathrm{tr}}(X)$. Bloch–Beilinson co-generation conjecture. There exists a birational idempotent $\varpi_i^X\in\operatorname{End}(\mathfrak{h}^{\circ}(X))$ with $(\varpi_i^X)^*\mathrm{H}^{*}_{\mathrm{tr}}(X)=\mathrm{H}^{i}_{\mathrm{tr}}(X)$, and for every such idempotent the co-algebra object $\mathfrak{h}^{\circ}(X)$ is co-generated by $\mathfrak{h}^{\circ}_i(X):=(X,\varpi_i^X)$, meaning that the co-induced morphism

$$\mathfrak{h}^{\circ}(X)\longrightarrow\operatorname{Sym}^*\mathfrak{h}^{\circ}_i(X)$$

is split injective. This is presented as a general expectation from Bloch–Beilinson philosophy and is open in general.