The small-cube conjecture for transitive sets

Let $X$ be a finite transitive subset of the unit sphere in $\mathbb{R}^d$ or $\mathbb{C}^d$. A unitary basis $L$ is a basis whose vectors are orthonormal with respect to the relevant real or complex inner product. Write $\langle \mathbf{v},\mathbf{x}\rangle$ for the inner product.

Small-cube conjecture. There is a unitary basis $L$ such that

$$\sup_{\mathbf{x}\in X,\mathbf{v}\in L}|\langle \mathbf{v},\mathbf{x}\rangle|\lesssim \frac{1}{\sqrt{\log d}}.$$

This conjecture proposes a coordinate system in which every point of every finite transitive set has uniformly small coordinates. The source presents it as a closely related conjecture; no resolution is supplied.