= Solution
The independent coordinates make the logarithm of the <probability density> ratio a sum. For coordinate $i$, its contribution to the <Kullback-Leibler divergence> is
$$
\mathbb E_q\left[\log\frac{\sigma_i}{\phi}-\frac{\theta_i^2}{2\phi^2}+\frac{\theta_i^2}{2\sigma_i^2}\right]=\log\frac{\sigma_i}{\phi}-\frac12+\frac{\phi^2}{2\sigma_i^2},
$$
because $\mathbb E_q\theta_i^2=\phi^2$. Thus, retaining all constants,
$$
\boxed{\mathcal J(\phi)=\frac{\phi^2}{2}\left(\frac1{\sigma_1^2}+\frac1{\sigma_2^2}\right)-1+\log\frac{\sigma_1\sigma_2}{\phi^2}}.
$$
Here $\sigma_1,\sigma_2>0$ and $\phi>0$, so both <normal densities> are nondegenerate.
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