The Structural Physical Approximation Conjecture
It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.
Shultz, Fred. The structural physical approximation conjecture, J. Math. Phys. 57 (2016), 015218 Copyright (2015) American Institute of Physics. http://dx.doi.org/10.1063/1.4938226