Local Hartshorne conjecture for varieties covered by lines

Let $X\subset {\mathbb P}^N$ be covered by lines, let $x\in X$ be a general point, and let $\mathcal L_x\subset {\mathbb P}^{n-1}$ be the variety of lines through $x$. Let $T$ be the span of $\mathcal L_x$ in ${\mathbb P}^{n-1}$. Local Hartshorne conjecture. If

$$\dim(\mathcal L_x)\geqslant \frac{n-1}{2}$$

and

$$\dim(\mathcal L_x)>2\operatorname{codim}(\mathcal L_x,T),$$

then $\mathcal L_x\subset {\mathbb P}^{n-1}$ is a complete intersection. This is proposed as a weaker, local version of Hartshorne's conjecture for Fano manifolds covered by lines; the source does not give a resolution.