= Solution
The favoured <weak charged current> transition is $\bar c\to\bar sW^-$ followed by $W^-\to d\bar u$. The <spectator quark> $u$ combines with $\bar u$ to form the $u\bar u/\sqrt2$ component of the <neutral pion>, while $d\bar s$ forms $K^0$. The product of <CKM matrix> factors is $V_{cs}V_{ud}^*$. The doubly <Cabibbo suppressed> transition is $\bar c\to\bar dW^-$ followed by $W^-\to s\bar u$, with factors $V_{cd}V_{us}^*$; the final $s\bar d$ is the <neutral antikaon>. Both are colour-suppressed internal-emission <Feynman diagrams>, with the same spectator routing.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2022-charm-decays.png]
{title=Favoured and doubly suppressed neutral-charm decay diagrams}
In the two-generation approximation to the <CKM matrix>, $V_{ud}=V_{cs}=\cos\theta_C$, $V_{us}=\sin\theta_C$ and $V_{cd}=-\sin\theta_C$. For equal magnitudes of the <hadronic matrix elements>,
$$
\boxed{\frac{|A(\bar D^0\to\bar K^0\pi^0)|}{|A(\bar D^0\to K^0\pi^0)|}=\tan^2\theta_C,\qquad
\frac{\Gamma_{\rm suppressed}}{\Gamma_{\rm favoured}}\simeq\tan^4\theta_C.}
$$
The rate ratio is about $2.8\times10^{-3}$ with the supplied <CKM matrix> magnitudes. The minus sign of a particular product of <CKM matrix> elements must be combined with the phases of the <hadronic matrix elements> and <kaon> basis states; it is not by itself the observable interference sign.
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