We have a sequence of discrete random variables $\boldsymbol x = \{x_1,\dots,x_n\} \in \mathcal X^n$ and a discrete conditional $P_{Y|X}$, i.e.
$$p(\boldsymbol y | \boldsymbol x) = \prod_i p(y_i|x_i). $$
If we assume that for all $p(\boldsymbol y | \boldsymbol x)>0$, we have $p(\boldsymbol y | \boldsymbol x) = f(x_1,\dots, x_n)g(y_1,\dots,y_n)$ which roughly means that we can't infer anything about $\boldsymbol x$ based on $\boldsymbol y$ besides the fact that $p(\boldsymbol y | \boldsymbol x)>0$.
I'm trying to come up with an example of a memoryless transformation such that that whenever $p(\boldsymbol y | \boldsymbol x) >0$ we have
$$\prod_i p(y_i|x_i) = f(\boldsymbol x)g(\boldsymbol y),$$
but $p(y_i|x_i) = h(y_i,x_i)$. That is, all possible input sequences $\boldsymbol x$ are equally likely given $\boldsymbol y$, but for some element pair $(x_i,y_i)$ we have $p(x_i|y_i)> p(x_i'|y_i)$.
Apologies if this is ill defined as stated, will try to clear up if more information is needed.
EDIT: Actually, what we can say is that whenever $p(\boldsymbol y | \boldsymbol x) >0$ we have
$$\prod_i p(y_i|x_i) = f(\boldsymbol y).$$
What we want to know is if this implies $p(y_i|x_i) = g(x_i)h(y_i)$.