Orthogonal tangent intersections characterize quadrics

Let $M\subset\mathbb R^n$, $n\geq 3$, be an open patch of a hypersurface of class $C^2$, and let $\Gamma_1,\Gamma_2$ be space curves. Orthogonal-intersection conjecture. If for every $p\in M$ there are unique intersection points $u_1=\Gamma_1\cap T_pM$ and $u_2=\Gamma_2\cap T_pM$ of the tangent hyperplane $T_pM$ with $\Gamma_1$ and $\Gamma_2$, respectively, and the vectors $u_1-p,u_2-p\in T_pM$ are orthogonal with respect to the second fundamental form $\mathrm{II}_p$ of $M$ at $p$, then $M$ is contained in a quadric. The statement is proposed as a far-reaching local generalization of the paper's main theorem and remains open.