Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality $$\frac{H(X_n\mid Y_n)}{\log n} \to 0,$$ which can be written $$\frac{H(X_n\mid Y_n)}{H(X_n)} \to 0 \qquad (\ast)$$ in the particular case when $X_n$ is uniformly distributed on $A_n$.

As shown by @AnthonyQuas here, $(\ast)$ fails in general. Now I'd like to know whether $$\limsup \frac{H(X_n\mid Y_n)}{H(X_n)} <1 \qquad (\ast\ast)$$ holds true in general, in the case when $H(X_n)\to \infty$.