The motivic multiple multiplicativity defect bound

Let $X$ be a smooth connected projective variety of dimension $n$ equipped with a CK decomposition. For an integer $m\geq 2$, an $m$-fold $r$-multiplicative CK decomposition is one whose $(m+1)$-fold intersection product respects the CK grading up to defect $r$; denote the smallest such defect by $\mathfrak{d}_m(X)$. The motivic multiplicativity defect conjecture. Any CK decomposition of $X$ is $m$-fold $mn$-multiplicative for every integer $m\geq 2$. In particular,

$$\mathfrak{d}_m(X)\leq mn.$$

This predicts a sharp linear bound on the obstruction to compatibility between multiple intersection products and CK decompositions; the paper presents it as a first step toward determining these defects, and no resolution is supplied.