Three-dimensional density of singular hyperbolicity

Let $M$ be a three-dimensional manifold, let ${\cal X}^1(M)$ denote the space of $C^1$ vector fields on $M$, and call a vector field singular hyperbolic when its chain-recurrent set is a finite union of singular-hyperbolic sets. Density conjecture. If $\operatorname{dim}(M)=3$, any vector field can be approximated in ${\cal X}^1(M)$ by singular hyperbolic ones. This is presented as a $C^1$ analogue of Smale's conjecture for surface diffeomorphisms; even for nonsingular vector fields, the source says it is open and interprets it as asserting density of hyperbolicity.