Corti–Hanamura's motivic decomposition conjecture

Let $f:X\to Y$ be a proper morphism between nonsingular varieties. A motivic decomposition is an isomorphism of relative Chow motives whose sheaf-theoretic specialization gives the decomposition of $Rf_*\mathbb{Q}_X$ into its perverse cohomology summands, equivalently an orthogonal decomposition of the relative diagonal class.

Corti–Hanamura's motivic decomposition conjecture. Every such morphism admits a motivic decomposition.

This conjecture is a motivic strengthening of the decomposition theorem and predicts that the perverse decomposition can always be lifted to relative Chow motives. Its resolution is not indicated in the supplied text.