convergence of discrete random variables with finite entropy

Question

Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ and $X_n \in Z$, is there a link between $\Pr(X_n \neq X) \to 0$ and $d(X_n,X) \to 0$ ?