The Uryson width conjecture for metrics on spheres

Let $m$ and $k$ be as in the claim, let $1\leq c\leq m$, and let $g_0$ denote the unit sphere metric on $S^m$. For a metric $g$ on $S^m$, write $\Lambda^k g\leq\Lambda^k g_0$ for the comparison of the induced metrics on $k$-vectors, and let $UW_{m-c}(S^m,g)$ denote the $(m-c)$-dimensional Uryson width. Uryson width conjecture. If

$$\Lambda^k g\leq\Lambda^k g_0\quad\text{and}\quad k\leq\frac{m}{c},$$

then

$$UW_{m-c}(S^m,g)\leq C(m).$$

The conjecture is known for $c=1$ and is trivial for $c=m$; it remains open for $2\leq c\leq m-1$.