Given random variables $X_1,...,X_4,Y\in\{0,1\}$ such that $X_1$ and $X_2$ are not independent. Do $X_1,...,X_4$ independent given $Y$? i.e. is $$ P[X=x|Y=y]=\Pi_{i=1}^4P[X_i=x_i|Y=y] \\ \forall x=(x_1,...,x_4)\in\{0,1\}^4,y\in\{0,1\}$$ where $X=(X_1,...,X_4)$?
Attempt:
There exists some $x\in\{0,1\}^4$ s.t. $P[X_1=x_1,X_2=x_2]\ne P[X_1=x_1]\cdot P[X_2=x_2]$. But I don't have data about the probabiliity of $[X_1=x_1,X_2=x_2,Y=y]$.