de Jong–Debarre conjecture on lines in smooth hypersurfaces

Let $X\subset\mathbb{P}^n$ be a smooth hypersurface of degree $d$, and let $F_1(X)\subset\mathbb{G}(1,n)$ denote the Hilbert scheme of lines contained in $X$. The expected dimension of $F_1(X)$ is $2n-d-3$. de Jong–Debarre conjecture. If $d\leq n$, then $F_1(X)$ has dimension $2n-d-3$. The bound $n\geq d$ is known to be optimal. The conjecture is known for $d\leq 8$, while the general case remains open.