The paving conjecture for zero-diagonal operators

Let $ell_2^n$ be the $n$-dimensional Hilbert space, let $T$ be a linear operator on $ell_2^n$ whose matrix has zero diagonal, and for $A\triangleleft\{1,\ldots,n\}$ let $Q_A$ denote the orthogonal projection onto the coordinates indexed by $A$. A partition $cA=\{A_j\}_{j=1}^r$ of cbrace{1,\ldots,n\} is called a paving of $T$ if it has $r$ parts. The paving conjecture. For every $\epsilon>0$, there exists a natural number $r$ such that, for every natural number $n$ and every such operator $T$, there is a paving $cA=\{A_j\}_{j=1}^r$ satisfying

$$\|Q_{A_j}TQ_{A_j}\|\leqepsilon\|T\|\qquad\text{for all }j=1,\ldots,r.$$

This conjecture is equivalent to the Kadison–Singer problem and concerns uniform blockwise smallness of zero-diagonal operators. The supplied source does not establish its resolution, so its status is recorded as open.