Motivic t-structure conjecture for rational geometric motives

Let $k$ be a field and let $\mu$ be a $t$-structure on the stable $\infty$-category $\textup{DM}_{\textup{gm}}(k;\mathbb{Q})$ of rational geometric Voevodsky motives. Its heart is $\textup{DM}_{\textup{gm}}(k;\mathbb{Q})^{\heartsuit}$, and $^{\mu}H^q$ denotes its cohomology functors. Let $\textup{Num}(k;\mathbb{Q})$ be the abelian category of rational numerical motives.

Motivic $t$-structure conjecture. There is a motivic $t$-structure on $\textup{DM}_{\textup{gm}}(k;\mathbb{Q})$ whose heart has semisimple part $\textup{Num}(k;\mathbb{Q})$, every motive has a filtration by rational numerical motives, and for every smooth projective $k$-variety $X$ each $^{\mu}H^qM(X)_{\mathbb{Q}}$ is semisimple in the heart. Moreover, there is a unique increasing weight filtration $W_{\bullet}M$ for every $M\in\textup{DM}_{\textup{gm}}(k;\mathbb{Q})$ such that, for each irreducible $P$, $W_mP=P$ and $W_{m-1}P=0$ whenever $P$ occurs in some $^{\mu}H^iM(X)_{\mathbb{Q}}(a)$ with $m=i-2a$ and $X$ any smooth projective $k$-variety.

This is presented as a notoriously difficult central conjecture in the theory of motives. The source uses it conditionally to deduce rational-motive invariance for K-equivalent smooth projective varieties; no resolution is supplied.