The Werner-state nondistillability conjecture

Fix $d\ge3$, let $F$ be the flip operator on a bipartite Hilbert space of local dimension $d$, and define the Werner states

$$\rho_W(t)=1-tF.$$

For $t\in(1/d,1/2]$, these states are NPT. A state is distillable if it is $k$-distillable for some integer $k\ge1$, where $k$-distillability means that a Schmidt-rank-at-most-two vector gives a negative expectation value for the $k$-fold partial transpose.

Werner-state nondistillability conjecture. None of the Werner states $\rho_W(t)$, $t\in(1/d,1/2]$, is distillable.

This is presented as a stronger form of the NPT bound-entanglement conjecture. The source gives no resolution; the conjecture concerns the remaining NPT interval after the known characterization of separability, NPT, and 1-distillability for Werner states.