The unrestricted cylindrical width conjecture for finite transitive subsets of complex spheres

Let $1\le k\le d$. Let $X$ be a finite transitive subset of the unit sphere in $\mathbb{C}^d$. Unrestricted cylindrical width conjecture. There is a complex $k$-dimensional subspace $W$ such that

$$\sup_{\mathbf{x}\in X}\lVert\operatorname{proj}_W\mathbf{x}\rVert_2 \lesssim \frac{1}{\sqrt{\log(2d/k)}}.$$

The preceding theorem proves this estimate under the additional restriction $k\le d/(\log(3d))^C$; the authors suspect that this restriction is unnecessary, so the conjecture concerns all $1\le k\le d$.