The structural physical approximation conjecture for optimal entanglement witnesses

Let $W$ be a normalized entanglement witness with ${\rm Tr}\,W=1$, and define

$$\widetilde{W}(p)=\frac{1-p}{n^{2}}\,\mathbb{I}_{n}\otimes\mathbb{I}_{n}+pW.$$

The structural physical approximation (SPA) of $W$ is an operator $\widetilde{W}(p)$ satisfying $\widetilde{W}(p)\geq 0$. Let $p_*$ be maximal such that $\widetilde{W}(p)\geq 0$ for $p\in[0,p_*]$. The structural physical approximation conjecture. If $W$ is an optimal entanglement witness, then $\widetilde{W}(p_*)$ defines a separable state. This is a conjecture about the separability of the boundary SPA of every optimal entanglement witness; the paper presents it as an interesting conjecture and does not establish it in general.