Let $p$ and $q$ be two joint distributions of finite random variables $X$ and $Y$. Recall the definition of conditional KL divergence between $p$ and $q$ of $X$ conditioned on $Y$: $D_{KL}(q(X|Y)||p(X|Y)) = \sum_yq(y)D_{KL}(q(X|Y=y)||p(X|Y=y))|Y=y]$ (e.g., Cover and Thomas, Elements of Information Theory).

Is it true that $D_{KL}(q(X|Y)||p(X|Y)) \leq D_{KL}(q(X)||p(X))$, in analogy to the inequality for entropy $H(X|Y) \leq H(X)$ ?

Note: it may seem like the opposite inequality follows from the convexity of KL divergence, but this is clearly not the case.