Scholl's vanishing and non-negative weight conjecture for mixed motives

Let $F$ be a number field, let $\pazocal{M}\pazocal{M}_F$ be the conjectural category of mixed motives over $F$ with neutral object $\mathbbm{1}$, and let $M$ be a mixed motive over $F$. Its motivic cohomology is $\operatorname{RHom}_{\pazocal{M}\pazocal{M}_F}(\mathbbm{1},M)$, whose $i$th cohomology is $\operatorname{Ext}^i_{\pazocal{M}\pazocal{M}_F}(\mathbbm{1},M)$. The weights of $M$ are the breaks of its weight filtration.

Scholl's conjecture. We expect that:

1. For $i\notin\{0,1\}$, $\operatorname{Ext}^i_{\pazocal{M}\pazocal{M}_F}(\mathbbm{1},M)=0$.
2. If the weights of $M$ are non-negative, $\operatorname{Ext}^1_{\pazocal{M}\pazocal{M}_F}(\mathbbm{1},M)=0$.

These expected properties describe the concentration and vanishing of motivic cohomology in the conjectural mixed-motive category; their resolution depends on the development of that category.