The motivic t-structure conjecture for Voevodsky motives

Let $k$ be the base field, let $\mathcal{DM}_c(k)$ denote the triangulated category of constructible Voevodsky motives over $k$, and let the $\ell$-adic realisation be the relevant realisation functor from $\mathcal{DM}_c(k)$ to an $\ell$-adic derived category. A $t$-structure on $\mathcal{DM}_c(k)$ is non-degenerate if the intersection of its two shifted halves is zero. Motivic t-structure conjecture. There exists a non-degenerate $t$-structure on $\mathcal{DM}_c(k)$ compatible with the tensor product and such that the $\ell$-adic realisation is $t$-exact. This conjecture predicts a canonical homological structure on constructible motives compatible with their tensor product and detected by $\ell$-adic realisation; its resolution is not established in the source.