Landsberg–Tommasi extension conjecture for Fano schemes of lines

Let $K$ be an algebraically closed field of characteristic zero, let $X^{n-1}\subset \mathbb P^n$ be a hypersurface of degree $n-\lambda$, and let $B\subset \mathbb F(X)$ be an irreducible component of dimension $n-2$. Let $\mathcal I_B$ and $X_B$ be as above, and define $\widetilde{\mathcal C}_x$ as above. Landsberg–Tommasi extension conjecture. If

$$\operatorname{codim}(X_B,X)\geq\lambda$$

and $\mathcal C_x$ is reduced for general $x\in X_B$, then for every $x\in X_B$,

$$\widetilde{\mathcal C}_x\cap X_{\mathrm{sing}}\neq\varnothing.$$

This extends the preceding singular-locus condition to smaller degrees under a codimension hypothesis and a reducedness assumption. The supplied text presents it as the authors' extension, but gives no resolution status.