Does connected wedge imply distillable entanglement?

The Ryu-Takayanagi formula predicts that two boundary subsystems $A$ and $C$ can exhibit large mutual information $I(A:C)$ even when they are spatially disconnected on the boundary and separated by a buffer subsystem $B$, as long as $A$ and $C$ have connected entanglement wedge in the bulk. However, whether the reduced state $\rho_{AC}$ contains distillable EPR pairs has remained a longstanding open problem. In this work, we resolve this problem by showing that: i) there is no LO-distillable entanglement at leading order in $G_N$, suggesting the absence of bipartite entanglement in a holographic mixed state $\rho_{AC}$, and ii) one-shot, one-way LOCC-distillable entanglement is given at leading order by locally accessible information $J^W(A|C)$, which is related to the entanglement wedge cross section $E^W$ involving the (third) purifying system $B$ via $J^W(A|C) = S_A - E^W(A:B)$. Namely, we demonstrate that a connected entanglement wedge does not necessarily imply nonzero distillable entanglement in one-shot, one-way LOCC. We also show that entanglement of formation $E_{F}(A:C)$ is given by $E^W(A:C)$ at leading order in holography.