unitarized the $\pi^+\pi^-$, $\pi^0 \eta$ (and coupled channels pairs) but the $\bar{K}^0$ has been left as a spectator. In principle we should also look at the interaction of $\bar{K}^0
\pi^-$ which can lead to the $\kappa$ resonance [@ramonet], yet the topology of Fig. \[Fig:mesonmesonFSI\] (a) does not favor $s$-wave interaction of $\bar{K}^0 \pi^-$. And furthermore, the $\kappa$ can also come from a different topology of the diagrams than those considered in Fig. \[Fig:mesonmesonFSI\] (a) for instance producing a $\pi^+$ meson from the $c$ quark via direct conversions of $W$ into $\pi^+$ (see section IV, Fig. \[wexchange\] (A)). This is why the $\kappa$ is better seen in the $D^+ \to K^- \pi^+ \pi^+$ reaction, as discussed in Ref. [@dosreis]. We do not consider the $\pi K$ interaction leading to the $\kappa$, with the argument that the $\kappa$, being a very broad resonance in the $\pi K$ invariant mass, only contributes a smooth background below the $\pi^+ \pi^-$, or $\pi^0
\eta$ invariant mass distribution when one looks for the $f_0(980)$ or $a_0(980)$ signals and is taken into account in experimental analysis of these two latter resonances. In this sense, the diagram of Fig. \[Fig:feyn\] chosen and the interaction that we have considered 