Physical hyperbolic measure for quotient maps away from the sink basin

Let $X$ be the vector field under consideration and let $Y$ be a vector field $C^2$ close to $X$. Let $g_Y$ be the quotient map, let $\mu_Y$ denote a candidate physical measure, and let the critical region refer to the region associated with the singular set. Physical-measure conjecture. Similarly to the one-dimensional setting, $g_Y$ admits a physical hyperbolic measure $\mu_Y$ for all those vector fields $Y$ which are $C^2$ close to $X$ and for which the stable manifold of the sink does not contain the critical region. Moreover, the basin of $\mu_Y$ should be the complement of the stable manifold of the sink. The claim is motivated by expected hyperbolicity away from the critical region and the sink basin, but remains unproved in the stated generality.