Anderson's paving conjecture

Let $\varepsilon>0$. For every $n\times n$ Hermitian matrix $T$ with zero diagonal, a diagonal projection is an orthogonal projection represented by a diagonal matrix. Anderson's paving conjecture. There exists an $r\in\mathbb{N}$ such that there are diagonal projections $P_{1},\cdots,P_{r}$ satisfying

$$\sum_{i=1}^{r}P_{i}=I$$

and

$$\left\Vert P_{i}TP_{i}\right\Vert \leq\varepsilon\left\Vert T\right\Vert,\quad\text{for }i=1,\ldots,r.$$

This is a matrix-paving formulation of the Kadison–Singer problem. The supplied text presents it as Anderson's original paving conjecture but does not state its resolution, so its database status remains open.