Matrix Spencer conjecture

Let $n$ be a positive integer, and let $A_1,\dots,A_n$ be self-adjoint $n\times n$ matrices, each with spectral norm at most one.

Matrix Spencer conjecture. There exists a universal constant $C$ such that

\min_{\varepsilon\in\\{\pm1\\}^n} \left\\| \sum_{k=1}^n \varepsilon_k A_k \right\\| \leq C\sqrt{n}.

This is a matrix-valued generalization of Spencer's discrepancy theorem. The source states that the conjecture was posed in earlier work and that the present paper proves it, so the claim is solved.