Variance-sensitive Matrix Spencer conjecture

Let $A_1,\ldots,A_n\in\mathbb R^{n\times n}$ be symmetric matrices satisfying $\|A_i\|_{\mathrm{op}}\le 1$. A coloring is a vector $x\in\{\pm1\}^n$. Variance-sensitive Matrix Spencer conjecture. There exists a universal constant $C$ such that every such family admits a coloring satisfying

$$\Bigl\|\sum_{i=1}^n x_iA_i\Bigr\|_{\mathrm{op}}\le C\Bigl\|\sum_{i=1}^n A_i^2\Bigr\|_{\mathrm{op}}^{1/2}.$$

This conjecture is motivated by variance-sensitive noncommutative Khintchine inequalities and would refine the dimension-dependent Matrix Spencer bound. The supplied text presents it as a conjecture and gives no resolution; its general status is therefore open.