Upper semicontinuity conjecture for singular physical measures

Let $\mathcal{V}$ be the open family of $C^2$ vector fields admitting a singular-hyperbolic attracting set $\Lambda$. For each $G\in\mathcal{V}$, let $s(G)\in\mathbb{Z}_0^+$ be the number of ergodic physical measures supported in $\Lambda$ and containing some equilibria. Upper semicontinuity conjecture. The function $s:\mathcal{V}\to\mathbb{Z}_0^+$ is upper semicontinuous. This conjecture concerns the stability of the number of physical measures whose supports contain singularities under perturbation; the paper notes that distinct physical measures may fuse under arbitrarily small perturbations.