Burago–Zalgaller–Ros conjecture on isoperimetric regions in projective spaces

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $\mathbb{P}^{n+1}(\mathbb{K})$ be the corresponding projective space. A projective subspace $\mathbb{P}^k(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$ has tubular neighborhoods in $\mathbb{P}^{n+1}(\mathbb{K})$. Burago–Zalgaller–Ros conjecture. The isoperimetric regions in the projective spaces $\mathbb{P}^{n+1}(\mathbb{K})$ are tubular neighborhoods of projective subspaces $\mathbb{P}^k(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$. This conjecture proposes the natural projective analogue of the Euclidean, hyperbolic, and spherical isoperimetric descriptions. The source notes that its real case was posed by Burago and Zalgaller in 1988; its general status is not established here.