Plane containment conjecture for low-degree curves on general hypersurfaces

Let $X \subset \mathbb{P}^{n+1}$ be a general hypersurface of degree $d$ and dimension $n \geq 3$, and let $C \subset X$ be a curve of degree $\delta$. Assume that

$$\frac{3}{2}n+2 \leq d \leq 2n-1.$$

Low-degree curve plane-containment conjecture. Every curve $C$ of degree $\delta \leq d$ must be contained in a $2$-plane. This extends the stated minimality theorem for plane curves to the indicated range of hypersurface degrees. The status of the conjecture is not resolved in the supplied source context.