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.