$\mathbb{E}_{x \sim p_{\text{data}}} \left[ \text{KL}(q_{\phi}(z|x) || p_{\theta}(z|x)) \right] = 0$ iff $p_\theta = p_{\text{data}}$

Question

Let $p_{\text{data}}$ be the data distribution given and assume $p_\theta, q_\phi$ be parametrized families of distributions.

Claim. $p_\theta(x) = p_{\text{data}}(x)$ for all $x$ iff $$ \mathbb{E}_{x \sim p_{\text{data}}} \left[ \text{KL}(q_{\phi}(z|x) || p_{\theta}(z|x)) \right] = 0. $$

Is this claim true and how can I prove it? I tried it myself but it seems like ($\Leftarrow$) part needs additional assumptions.