Mutual information of the joint distribution and the marginal distribution if two random variables are independent?

Question

if $Y \rightarrow X$ and $Y \rightarrow Z$, is it possible to find the equality/inequality between $$ I(X,Z; Y) $$ and $$ I(X;Y) + I(Z;Y) $$

Or equivently, given $I(X,Z;Y)= I(X; Y) + I (Z; Y |X)$, what's the relationship between $$ I(Z; Y|X) $$ and $$ I(Z; Y) $$