= Solution
Use the explicit charged channels $D^+\to\pi^+\pi^+\pi^-$ and $K^+\to\pi^+\pi^+\pi^-$, labeling the two positive <pions> as 1,2 and the negative one as 3. With $m_\pi=0.139\,\mathrm{GeV}$, the opposite-charge pair <mass> squares have the ranges
$$
\boxed{4m_\pi^2\le s_{13},s_{23}\le(M_0-m_\pi)^2.}
$$
Their exact correlated boundaries follow by taking the pair rest-frame angle to its endpoints. For general daughter <masses>, define $\Lambda(a,b,c)=a^2+b^2+c^2-2ab-2ac-2bc$. At fixed $s=s_{13}$,
$$
\boxed{s_{23}^{\pm}=m_2^2+m_3^2+
\frac{(M_0^2-m_2^2-s)(s+m_3^2-m_1^2)\pm\sqrt{\Lambda(M_0^2,s,m_2^2)\Lambda(s,m_1^2,m_3^2)}}{2s}.}
$$
The charged-pion thresholds give the following numerical comparison:
|| Parent | $Q=M_0-3m_\pi$ | Allowed single-axis interval, $\mathrm{GeV}^2$ | Dalitz area, $\mathrm{GeV}^4$ \\
| $D^+$ | $1.453\,\mathrm{GeV}$ | $0.07728$ to $2.99636$ | approximately $5.074$ \\
| $K^+$ | $0.077\,\mathrm{GeV}$ | $0.07728$ to $0.12603$ | approximately $0.001611$
The area is about $3150$ times smaller for the <kaon>. Including the $M_0^{-2}$ <Lorentz-invariant phase space>[phase-space] prefactor, its integrated labelled three-body <Lorentz-invariant phase space>[phase space] is about $220$ times smaller; neither comparison predicts a ratio of <Weak interaction>[weak decay] widths without the <matrix elements>.
A $\rho^0\to\pi^+\pi^-$ intermediate state produces horizontal or vertical bands near $s_{13}$ or $s_{23}=m_\rho^2\simeq(0.775\,\mathrm{GeV})^2$ in the <Charm quark>[charm] decay. <Partial wave>[Scalar pion-pair amplitudes], including broad low-mass $S$-wave contributions and accessible $f_0(980)$, can also populate the plot. The <kaon> pair <mass> cannot exceed $0.355\,\mathrm{GeV}$, so it cannot contain an <On shell>[on-shell] $\rho^0$ or $f_0(980)$ band; smooth nonresonant and rescattering effects remain possible. Channels containing neutral <pions> use their supplied $0.135\,\mathrm{GeV}$ <mass> and shift these boundaries slightly without changing the conclusion. The figure shows kinematic domains and representative accessible bands, not an assumed measured intensity.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2017-dalitz-domains-and-resonances.png]
{title=Calculated charged-three-pion Dalitz domains and accessible resonance bands}
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