Conjecture on bounded second fundamental form and spherical topology

Let $M$ be the immersed hypersurface satisfying the hypotheses of the Main Theorem, let $B$ be its second fundamental form, and for $q>0$ define the normalized $L^q$ norm

$$\|B\|_q=\left(\frac{1}{|M|}\int_M |B|^q\right)^{1/q}.$$

Spherical embedding conjecture. If, in addition, $M$ satisfies

$$|M|^{\frac{1}{n}}\|B\|_q\leq A$$

for some $q>0$, then $M$ is embedded and diffeomorphic to a round sphere.

The conjecture proposes that a normalized integral bound on the second fundamental form prevents the hypersurface in the Main Theorem from being highly twisted and forces spherical topology. Its status is open, even for hypersurfaces in a space form.