Example: Dropping uniformity changes even the coordinate value

Let $S$ be a uniform sign, but prepare

rather than a uniform point of ${\Omega }_2$. Apply independent crossover-$1/4$ noise to both coordinates, and take $f(X)=X_1=S$. Then

Indeed, $Y_2$ carries nonzero additional information about $S$ conditional on $Y_1$: both conditional values of $S$ have positive probability, and their $Y_2$-laws differ. Strict concavity of binary entropy makes the information increase strict. Explicitly,

The two observations disagree with probability $3/8$, in which case the posterior is fair; when they agree, the posterior error is $1/10$.

Thus the frozen universal constant cannot simply be retained after dropping the uniform-product-input assumption.