A reduction of the separability problem to SPC states in the filter normal form

It was recently suggested that a solution to the separability problem for states that remain positive under partial transpose composed with realignment (the so-called symmetric with positive coefficients states or simply SPC states) could shed light on entanglement in general. Here we show that such a solution would solve the problem completely. Given a state in $\mathcal{M}_k\otimes\mathcal{M}_m$, we build a SPC state in $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$ with the same Schmidt number. It is known that this type of state can be put in the filter normal form retaining its type. A solution to the separability problem in $\mathcal{M}_k\otimes\mathcal{M}_m$ could be obtained by solving the same problem for SPC states in the filter normal form within $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$. This SPC state can be built arbitrarily close to the projection on the symmetric subspace of $\mathbb{C}^{k+m}\otimes\mathbb{C}^{k+m}$. All the information required to understand entanglement in $\mathcal{M}_s\otimes\mathcal{M}_t$ $(s+t\leq k+m)$ lies inside an arbitrarily small ball around that projection. We also show that the Schmidt number of any state $\gamma\in\mathcal{M}_n\otimes\mathcal{M}_n$ which commutes with the flip operator and lies inside a small ball around that projection cannot exceed $\lfloor\frac{n}{2}\rfloor$.