Smooth 4-dimensional Schoenflies problem

Let $S^4=\{x\in\mathbb R^5:\|x\|=1\}$ with its standard smooth structure and $D^4$ the closed unit ball in $\mathbb R^4$. Let $M\subset S^4$ be a smooth $3$-dimensional submanifold diffeomorphic to $S^3$, given as the image of a smooth embedding $f:S^3\to S^4$ (a $C^\infty$ injective immersion). By the Jordan–Brouwer separation theorem $M$ separates $S^4$ into two compact connected $4$-manifolds with common boundary $M$. The conjecture: for every $x\in S^4\setminus M$, with $C_x$ the component of $S^4\setminus M$ containing $x$,

i.e. $M$ is unknotted.

This is the only dimension in which the smooth Schoenflies problem is open; a negative answer would give an exotic smooth $D^4$ embedded in $S^4$.

Formalization Note 'Diffeomorphic to $D^4$' is IsSmoothClosedBallIn 3: the image of the closed unit ball under a $C^\infty$ map $\mathbb R^4\to S^4$ that is injective and has injective differential at every point of the closed ball (boundary included). The conclusion is required for both complementary components.