Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square whose top row is $L_{E\vee F}(X)\to L_E(X)$, and whose lower row is $L_F(X)\to L_{E\wedge F}(X)$? If not, is it known what conditions I need to place on $E$ and $F$ to make this all work out? Does anyone know if I can iterate this process over some set of homology theories?

Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square whose top row is $L_{E\vee F}(X)\to L_E(X)$, and whose lower row is $L_F(X)\to L_{E\wedge F}(X)$? If not, is it known what conditions I need to place on $E$ and $F$ to make this all work out? Does anyone know if I can iterate this process over some set of homology theories?

I went ahead and made this a reference request, because I imagine it could a rather significant answer.