The Esnault–Srinivas–Viehweg converse for Chow and cohomological products

Let $X$ be a smooth complete variety of dimension $n$ over $\mathbb{C}$, and let $n=n_1+\cdots+n_r$ be a partition with $n_i\in\mathbb{N}_{>0}$. Property (P1) means that there exists a Zariski open $V\subset X$ such that intersection product induces a surjection

$$A^{n_1}V_{\mathbb{Q}}\otimes A^{n_2}V_{\mathbb{Q}}\otimes\cdots\otimes A^{n_r}V_{\mathbb{Q}}\to A^nV_{\mathbb{Q}}.$$

Property (P2) means that there exists a Zariski open $V\subset X$ such that cup product induces a surjection

$$H^{n_1}(V,\mathbb{Q})\otimes H^{n_2}(V,\mathbb{Q})\otimes\cdots\otimes H^{n_r}(V,\mathbb{Q})\to H^n(V,\mathbb{Q})/N^1,$$

where $N^\ast$ denotes the coniveau filtration. Esnault–Srinivas–Viehweg's converse. If (P2) holds, then (P1) holds. This is the conjectural converse to the known implication from (P1) to (P3) and from (P2) to (P3); it is related to the generalized Hodge and Bloch–Beilinson conjectures, and is not established unconditionally in the source.