= Solution
Take $|0\rangle$ to have $\sigma_z=+1$ and use angular-frequency <gyromagnetic ratios>. Neglecting interspin coupling, the <Hamiltonian> of the <Zeeman energy> is
$$
\boxed{H=-\frac{\hbar B}{2}(\gamma_1 Z_1+\gamma_2 Z_2).}
$$
For $\gamma_1=\gamma_2=\gamma$, define $\epsilon=\hbar\gamma B/(k_BT)$. The exact thermal <density matrix> in the order $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ is
$$
\rho_{\mathrm{th}}=\frac{\operatorname{diag}(e^\epsilon,1,1,e^{-\epsilon})}{4\cosh^2(\epsilon/2)}.
$$
At high <temperature>, $|\epsilon|\ll1$, expand both numerator and normalization to obtain
$$
\boxed{\rho_{\mathrm{th}}=\frac14\operatorname{diag}(1+\epsilon,1,1,1-\epsilon)+O(\epsilon^2),\qquad \epsilon=\frac{\hbar\gamma B}{k_BT}.}
$$
For positive $\gamma$, $|00\rangle$ is the most populated state; a negative <gyromagnetic ratio> reverses that population preference under this basis convention.
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