The injectivity conjecture for polynomial cycles on powers of a K3 surface

Let $S$ be an algebraic $K3$ surface, let $m$ be any integer, and let $o$ denote the distinguished zero-cycle used in the construction. For $P\in CH(S^m)$, assume that $P$ is a polynomial expression in

$$\operatorname{pr}_i^*c_1(L_s),\quad L_s\in \operatorname{Pic} S,\quad \operatorname{pr}_j^*o,\quad \operatorname{pr}_{kl}^*\Delta_S.$$

Polynomial-cycle injectivity conjecture. If the cohomology class of $P$ vanishes, then $P$ vanishes in the Chow group:

$$[P]=0\quad\Longrightarrow\quad P=0.$$

The statement is used as an auxiliary conjectural framework in the paper, which proves particular cases sufficient for its main results. Its general validity is not established here.