Barbosa–Viana pinching conjecture in the four-dimensional ball

Let $\Sigma^3\subset B^4\subset\mathbb{R}^4$ be a capillary minimal hypersurface in the unit ball; the free boundary case corresponds to contact angle $\pi/2$. Assume that

$$|x^\perp|^2\,|A(x)|^2 \le \frac{3}{2}$$

for all $x\in\Sigma$. Barbosa–Viana conjecture. One of the following should hold: (i) $\Sigma$ is congruent to the equatorial three-disk $\mathbb{D}^3$; or (ii) $\Sigma$ is congruent to an $O(2)\times O(2)$-invariant free boundary minimal hypersurface, with such examples topologically homeomorphic to the solid torus $\mathbb{D}^2\times\mathbb{S}^1$. This is a higher-dimensional analogue of rigidity results for free boundary minimal surfaces, where current results provide topological classification but not a complete identification of geometric models. The concrete uniqueness and existence problems in this higher-dimensional setting remain unresolved.