Hartshorne conjecture for varieties of lines

Let $X\subset\mathbb{P}^N$ be a smooth irreducible non-degenerate variety of dimension $n$ covered by lines. For a general point $x\in X$, let $\mathcal{L}_x\subset\mathbb{P}^{n-1}$ denote the variety of lines through $x$, and let $T=\langle\mathcal{L}_x\rangle\subseteq\mathbb{P}^{n-1}$. Hartshorne conjecture for varieties of lines. If $\dim(\mathcal{L}_x)\geqslant\frac{n-1}{2}$ and $\dim(\mathcal{L}_x)>2\operatorname{codim}_T(\mathcal{L}_x)$, then $\mathcal{L}_x\subset\mathbb{P}^{n-1}$ is a complete intersection. The conjecture is a proposed reduction of the Fano Hartshorne problem to the geometry of varieties of lines; the source gives some classes in which it is known, but does not resolve it in general.