Generic no-cut-and-paste conjecture for Lorenz maps

Let $O$ be the collection of vector fields under consideration, let $\mathcal{L}$ be the collection of Lorenz maps that form factor maps for vector fields $V\in O$, and let $\mathcal{CP}$ be the collection of Lorenz maps that do not satisfy the general cut-and-paste condition. Generic no-cut-and-paste conjecture. There should be a topology on $\mathcal{L}$ such that $\mathcal{CP}$ is open and dense, the reduction theorem defines a surjective map

$$\mathcal{F}:O\to\mathcal{L},$$

matching each vector field with its corresponding Lorenz map, $\mathcal{F}$ is continuous for the $C^1$ topology on $O$ and this topology on $\mathcal{L}$, and

$$\mathcal{F}^{-1}(\mathcal{L}\setminus\mathcal{CP})$$

is precisely the collection of vector fields in $O$ generating a pair of homoclinic trajectories to the origin. The claim concerns the genericity of the no-cut-and-paste condition and the relation between its failure and homoclinic pairs; the source leaves the appropriate topology and the asserted correspondence unresolved.