Weak non-uniform sectional expansion conjecture for physical and SRB measures

Let $G$ be a $C^2$ vector field with a partially hyperbolic attracting set $\Lambda=\Lambda_G(U)$, and let $\Omega\subset U$ be a positive-volume subset on which the weak non-uniform sectional expansion condition holds. Weak sectional expansion conjecture. In the same setting as the stated theorems, this condition is enough to obtain the same conclusions concerning the existence and finiteness of physical/SRB measures. The analogous result is known for local diffeomorphisms, while the corresponding flow statement is presented as open.