Fundamental group of product of schemes Is the \'etale fundamental group of the product of two schemes $X_1$ and $X_2$ related to the \'etale fundamental groups of $X_1$ and $X_2$ individually?

The following is proven SGA 1, X.1.7: 

Let $k$ be an algebraically closed field, and $X,Y$ connected $k$-schemes such that $X$ is proper and $Y$ is locally noetherian. Let $x,y$ be geometric points of $X,Y$ with values in the same algebraically closed field. Then the canonical homomorphism of profinite groups $\pi_1^{ét}(X \times_k Y,(x,y)) \to \pi_1^{ét}(X,x) \times \pi_1^{ét}(Y,y)$ is an isomorphism.

We really need properness, since the isomorphism fails for $X=Y=\mathbb{A}^1_{\mathbb{F}_p}$ (see SGA 1, X.1.10).
