Flatness conjecture for stable nonlocal minimal hypercones in dimensions three through seven

Let $s\in(0,1)$, and consider stable $s$-minimal hypercones in $\mathbb{R}^n\setminus\{0\}$. Flatness conjecture. For every integer $n$ with $3\leqslant n\leqslant 7$, there exists $s_*\in(0,1)$ such that, for every $s\in(s_*,1)$, the only stable $s$-minimal hypercones in $\mathbb{R}^n\setminus\{0\}$ are hyperplanes.

This is the nonlocal analogue of the classical flatness result for stable minimal cones in dimensions at most seven, contrasted with the Simons-cone counterexample in dimension eight. Establishing this classification would, via the stated implication for stable hypersurfaces, yield flatness of stable $s$-minimal hypersurfaces of class $C^2$ in these dimensions for $s$ sufficiently close to one.