Bloch–Beilinson compatibility conjecture for multiple intersection products

Let $X$ be a smooth connected projective variety of dimension $n$ with CK projectors $\{\pi_i^X\}_{0\leq i\leq 2n}$, and let $\Delta_I^X\in\operatorname{CH}^{mn}(X^{m+1})$ be the $(m+1)$-fold small diagonal, where $m\geq 2$. For integers $i_1,\ldots,i_m$ and $k$, let $\pi_k^X\circ\Delta_I^X\circ(\pi_{i_1}^X\otimes\cdots\otimes\pi_{i_m}^X)$ denote the corresponding CK-graded component. Bloch–Beilinson's multiple-intersection conjecture. For each integer $k>\sum_{\ell=1}^{m}i_\ell$,

$$\pi_k^X\circ\Delta_I^X\circ(\pi_{i_1}^X\otimes\cdots\otimes\pi_{i_m}^X)=({}^t\pi_{i_1}^X\otimes\cdots\otimes{}^t\pi_{i_m}^X\otimes\pi_k^X)_*\Delta_I^X=0$$

in $\operatorname{CH}^{mn}(X^{(m+1)n})$. This is presented as the expected compatibility of the Bloch–Beilinson filtration with every multiple intersection product; the paper does not state a resolution.