Higher-dimensional Alexandrov isodiametric conjecture

Let $\Sigma$ be a closed oriented convex Riemannian manifold of dimension $d$, with Riemannian volume $A$ and intrinsic diameter $D$. Let $B^d$ denote the $d$-dimensional unit ball, and let $|B^d|$ be its volume.

Higher-dimensional Alexandrov conjecture. The inequality

$$\frac{A}{D^d} \leq \frac{|B^d|}{2^{d-1}}$$

holds for any such manifold.

This is proposed as the extension to arbitrary dimension of Alexandrov's surface conjecture. The inequality has been proved for closed spherically symmetric Riemannian manifolds diffeomorphic to $S^d$ that are isometrically embedded in $\mathbb{R}^{d+1}$, but remains open for general closed oriented convex Riemannian manifolds.