Local-to-global complete-intersection 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 $x$. Local-to-global complete-intersection conjecture. If $\mathcal L_x$ is a smooth, irreducible, non-degenerate complete intersection, then $X$ is a complete intersection. The source motivates this by observing that complete intersections have complete-intersection varieties of lines through a general point. The converse is proposed for varieties covered by lines and remains open in the source.