Essential Supremum and Supremum in anticipation


Suppose that $ {Z_i } _ {i in I} $ are a family of densities in $ L ^ 2 ( Omega, mathcal {F}, mathbb {P}) $, and $ X = L ^ 2 ( Omega, mathcal {F}, mathbb {P}) $, When is it true?
$$
sup_ {i in I} mathbb {E} left[Z_icdot
(X- mathbb{E}[Z_icdot X|mathcal{G}]) ^ 2
Law]=
sup_ {i in I} mathbb {E} left[Z_icdot
(X- operatorname{esssup}_{i in I}mathbb{E}[Z_icdot X|mathcal{G}]) ^ 2
Law]?
$$

I've seen similar questions, but I have not encountered anything like that here.