Gromov's waist comparison conjecture for positively curved Riemannian manifolds

Let $X$ be an $n$-dimensional Riemannian manifold with sectional curvature $K\geq 1$, and let $f:X\to\mathbb{R}^k$ be a continuous map. For a subset $Y\subset X$, write $Y+\varepsilon=\{x\in X\mid d(x,Y)\leq\varepsilon\}$. Let $\mathbb{S}^n$ denote the canonical Riemannian $n$-sphere with sectional curvature equal to $1$. Gromov's conjecture. There exists $z\in\mathbb{R}^k$ such that, for every $\varepsilon>0$,

$$\frac{\operatorname{vol}_n(f^{-1}(z)+\varepsilon)}{\operatorname{vol}_n(X)}\geq \frac{\operatorname{vol}_n(\mathbb{S}^{n-k}+\varepsilon)}{\operatorname{vol}_n(\mathbb{S}^n)}.$$

This is a waist comparison asserting that positively curved Riemannian manifolds have normalized fiber-neighborhood volume at least that of the unit round sphere. The source notes that the conjecture is proved when $X$ is itself the canonical Riemannian sphere; the general case, particularly for positive $\varepsilon$, remains open.