The chain-length characterization conjecture for Fano manifolds of Picard number one

Let $X$ be a Fano manifold of dimension $n$ and Picard number $1$. Let $\underline{N}_X$ and $\overline{N}_X$ denote the lower and upper lengths, respectively, of chains of higher-order minimal families of rational curves on $X$.

Chain-length characterization conjecture.

1. If

$$\underline{N}_X\ge \left\lceil\frac{n}{2}\right\rceil,$$

then $X$ is isomorphic to either $\mathbb{P}^n$ or the quadric $Q^n$. 2. If

$$\overline{N}_X\ge \left\lceil\frac{n}{2}\right\rceil,$$

then $X$ is isomorphic to one of $\mathbb{P}^n$, $Q^n$, and $G\left(2,\frac{n}{2}+2\right)$.

The bounds arise from explicit chains for projective spaces, quadrics, and Grassmannians. A special case has been proved, but the full conjecture remains open according to the supplied source.