Weaver's $KS_2$ conjecture

Let $d,m$ be positive integers, and let $w_1,\dots,w_m\in\mathbb{C}^d$ satisfy $\lVert w_i\rVert\leq 1$ for every $i$. Suppose there are universal constants $\eta\geq 2$ and $\theta>0$ such that

$$\sum_{i=1}^m |\langle u,w_i\rangle|^2=\eta$$

for every unit vector $u\in\mathbb{C}^d$. Weaver's $KS_2$ conjecture. There exists a partition $S_1,S_2$ of $\{1,\dots,m\}$ such that

$$\sum_{i\in S_j}|\langle u,w_i\rangle|^2\leq\eta-\theta$$

for every unit vector $u\in\mathbb{C}^d$ and each $j\in\{1,2\}$. This conjecture, due to Weaver, was shown to imply a positive solution to the Kadison--Singer problem; in the source it is followed by a stronger version obtained from the preceding partition theorem, and the stated conjecture is resolved by the paper's main result.