Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

Do there exist smooth morphisms of schemes $=X_i\rightarrow Y_i$ such that $W_i:Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and form an etale cover of $W$? [Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]