Generic $C^1$ bounds for physical and SRB measures

Consider generic $C^1$ vector fields having a sectional-hyperbolic attracting set. Let the bounds referred to in Theorem~ and Corollary~ be the corresponding bounds for the number of ergodic physical and SRB measures. Generic $C^1$ bounds conjecture. The number of ergodic physical/SRB measures satisfies the same bounds as those given by Theorem~ and Corollary~. Results for $C^1$ vector fields give partial motivation, but the generic assertion with the same bounds remains open in the supplied text.