For two $\sigma$-fields, $\mathcal{A}$ and $\mathcal{B},$ we have the notion of HGR maximal correlation

$$\rho(\mathcal{A},\mathcal{B}) = \sup \frac{Efg-Ef.Eg}{\sqrt{\mathsf{Var}(f).\mathsf{Var}(g)}}$$

where the supremum is over $f,g$ being $L^2$ functions measurable with respect to $\mathcal{A}$ and $\mathcal{B}$ respectively.

We also have the notion of correlation

$$\delta(\mathcal{A},\mathcal{B}) = \sup_{A\in\mathcal{A}, B\in\mathcal{B}} \frac{P(A\cap B)-P(A)P(B)}{\sqrt{P(A)P(A^c)P(B)P(B^c)}},$$

that is, the supremum over $f,g$ being indicator functions measurable with respect to $\mathcal{A}$ and $\mathcal{B}$ respectively.

Is it true that these two measures of correlation are identical when $\mathcal{A}$ and $\mathcal{B}$ both have finite number of sets?