Conjecture that non-singular hypersurfaces are not unions of maximal sub-boundary planes

Let $n\geq 5$, let $X\subset\mathbb{P}^n$ be a non-singular hypersurface of degree $d\geq 3$, and set

$$m=\lfloor(n-1)/2\rfloor.$$

An $m$-plane means a linear projective subspace of dimension $m$ contained in $X$. The non-union-of-planes conjecture. The hypersurface $X$ is not a union of $m$-planes.

The question concerns the possible dimension of the Fano scheme of $m$-planes in the sub-boundary case. The source says that resolving it would simplify the proof of a stated theorem. It also records that the conjecture was proved when $d\geq 4$ by Roya Beheshti, so only the remaining cases are open on the supplied evidence.