Generic transversality conjecture for nonlinear dynamics

Let $M$ be a compact manifold of dimension $m$, let $P:M\rightarrow\mathbb{R}^d$ be smooth, and let $\mathfrak{F}$ be the quasi-stratification of $M$ by the level sets $\mathfrak{F}_x$ of $P$. Let $N$ be a finite union of immersed submanifolds of dimension at most $n$, with transverse self-intersections, and let $D^{\infty}(M)$ denote the space of smooth diffeomorphisms from $M$ to itself. For $F\in D^{\infty}(M)$, define $I_x=F(N)\cap\mathfrak{F}_x$. Generic transversality conjecture. For an open and dense set of $F\in D^{\infty}(M)$,

$$\dim(I_x)\leq\max(n-d,0)\qquad\forall x\in\mathbb{R}^d,$$

and $I_x$ is a finite union of stably immersed submanifolds. This conjecture is intended to extend the preceding linear-dynamics argument to nonlinear dynamics; the source gives no resolution, and the genericity and stable-immersion assertions are the ingredients used to control successive intersections of projected dynamical images.