= Solution
Use the factorization $p(\theta,\mathcal D)=p(\theta\mid\mathcal D)p(\mathcal D)$ inside the <evidence lower bound>:
$$
\begin{aligned}
\mathcal L(\phi)&=\mathbb E_q\left[\log p(\mathcal D)+\log\frac{p(\theta\mid\mathcal D)}{q(\theta;\phi^2)}\right]\\
&=\log p(\mathcal D)-\operatorname{KL}\bigl(q(\theta;\phi^2)\Vert p(\theta\mid\mathcal D)\bigr).
\end{aligned}
$$
Hence
$$
\boxed{\log p(\mathcal D)=\mathcal L(\phi)+\mathcal J(\phi)}.
$$
The <marginal likelihood> is independent of the variational parameter $\phi$, so \b[maximizing the <evidence lower bound> is equivalent to minimizing the <Kullback-Leibler divergence>]. Nonnegativity of the <Kullback-Leibler divergence> also gives $\mathcal L(\phi)\leq\log p(\mathcal D)$.
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