Improved symmetry-rank conjecture for positive quaternionic Kähler manifolds of odd quaternionic dimension

Let $M$ be a positive quaternionic Kähler manifold of dimension $8m+4$, and let $\operatorname{rank}(M)$ denote the rank of its isometry group. Improved symmetry-rank conjecture. If

$$\operatorname{rank}(M)\ge m+3,$$

then $M$ is isometric to $\mathbb{H}P^{2m+1}$ or $\operatorname{Gr}_2(\mathbb{C}^{2m+3})$. This improves the lower-bound threshold in the preceding symmetry-rank theorem by $1$ when the quaternionic dimension is odd; the claim concerns the classification of positive quaternionic Kähler manifolds with sufficiently large isometry-group rank.