Berger's conjecture on the isoperimetric profile of projective spaces

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $I_{\mathbb{P}^{n+1}(\mathbb{K})}$ denote the isoperimetric profile of $\mathbb{P}^{n+1}(\mathbb{K})$, defined by

$$I_{\mathbb{P}^{n+1}(\mathbb{K})}(v)=\inf\{\lvert\partial\Omega\rvert:\Omega\subset\mathbb{P}^{n+1}(\mathbb{K}),\ \lvert\Omega\rvert=v\}.$$

A projective subspace $\mathbb{P}^k(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$ has successive tubular neighborhoods whose perimeters define a candidate profile. Berger's conjecture. The isoperimetric profile of $\mathbb{P}^{n+1}(\mathbb{K})$ is given by the perimeter of successive tubular neighborhoods of projective subspaces $\mathbb{P}^k(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$. This is a profile-level formulation of the proposed classification of isoperimetric regions in projective spaces. The source attributes this formulation to Berger and does not provide evidence of a resolution.