Sparse approximation conjecture for John decompositions

Let $u_1,\ldots,u_n\in S^{d-1}$ be unit vectors for which there exist positive numbers $\alpha_1,\ldots,\alpha_n$ satisfying

$$\sum_{i=1}^n \alpha_i u_i=0, \qquad \sum_{i=1}^n \alpha_i u_i\otimes u_i=I_d,$$

where $I_d$ is the identity operator on $\mathbb{R}^d$. Sparse John decomposition conjecture. There exists a subset $\sigma\subset[n]$ with $|\sigma|\leq 2d$ such that

$$B^d\subset cd\,\operatorname{conv}\{u_i:i\in\sigma\}$$

for an absolute constant $c>0$.

This conjecture seeks a linear-in-$d$ approximation of the Euclidean ball using at most $2d$ vectors from a John decomposition. The surrounding results establish related sparse approximation statements for polytopes, while the conjectured bound itself is presented as open and would imply an asymptotically sharp bound for the volume parameter $v(d)$.