= Solution
Let $|1\rangle=|D^0\rangle$, $|2\rangle=|\bar D^0\rangle$. In the rest frame, the <CPT theorem> interchanges these states up to irrelevant phases. For the weak <Hamiltonian operator> $H'$, with $H'=H'^{\dagger}$, and its antiunitary <CPT symmetry> operator $\Theta$,
$$
\langle\Theta1|H'|\Theta1\rangle=\langle1|H'|1\rangle^*.
$$
Its diagonal <matrix elements> are real by <Hermitian conjugation>, so
$$
\boxed{R_{11}=R_{22}.}
$$
In the specified phase convention, the <CP symmetry> operator on this two-state subspace is $-\sigma_x$. If the <weak interaction> preserves <CP symmetry>, $\sigma_xR\sigma_x=R$, giving
$$
\boxed{R_{12}=R_{21}.}
$$
Independently, <Hermitian conjugation> gives $R_{21}=R_{12}^*$ for this $H'$. Hence <CP symmetry> makes the off-diagonal element real in this convention; that conjugation relation alone does not imply <CP symmetry>.
For <neutral meson mixing> described by the decay-effective <Hamiltonian operator> $R=M-i\Gamma/2$, the effective matrix is generally not a <Hermitian matrix>. The <CPT constraints on neutral-meson mixing> still give equality of the complex diagonal entries, expressing equal masses and total <decay widths>, and <CP symmetry> gives equality of the off-diagonal entries in the specified convention. One must not additionally impose $R_{21}=R_{12}^*$ on this decay-effective matrix.
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