The local complete-intersection 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 $x$. Local complete-intersection conjecture. If $\mathcal{L}_x\subset\mathbb{P}^{n-1}$ is a smooth irreducible non-degenerate complete intersection, then $X$ is a complete intersection. The conjecture asserts a converse to the fact that the variety of lines of a complete intersection is itself a complete intersection; the source explicitly says this converse is believed but does not establish it.