On Fano manifolds of Picard number one with big automorphism groups

Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $dim mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $mathbb{P}^n, mathbb{Q}^n$ or ${rm Gr}(2,5)$. Furthermore, the equality $dim mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${rm Lag}(6)$ or a general hyperplane section of ${rm Gr}(2,5)$.