Relative phases in Dalitz plot amplitudes for $D^0 \to K_S π^+ π^-$ and $D^0 \to π^0 K^+ K^-$

Relative phases of amplitudes for $D$ meson decays to a light pseudoscalar meson $P$ and a light vector meson $V$ decaying to two pseudoscalar mesons will lead to characteristic interferences on the three-body Dalitz plot. These phases may be compared with predictions of a flavor-symmetric treatment which extracts contributing amplitudes and their relative phases from a fit to $D \to PV$ decay rates. Good agreement was found previously for the cases of $D^0 \to K^- π^+ π^0$ and $D^0 \to π^+ π^- π^0$. The present work is devoted to the decays $D^0 \to K_S π^+ π^-$ and $D^0 \to π^0 K^+ K^-$, for which agreement is not found. Several suggestions are offered for this discrepancy.