Total variation in product

Question

Let $a(x),b(x)$ be two density functions on $X$ and $c(y),d(y)$ be two density functions on $Y$, i.e. they are non-negative real functions such that $\int_X a=\int_X b=\int_Y c=\int_Y d=1$. Is it true that the following inequality holds? $$\int_X|a-b| \int_Y |c-d| \leq \left(\int_{X \times Y} |ac-bd|\right)^2$$ Thanks in advance!