Sarason's conjecture for the Bergman space

Let $A^2(\mathbb{D})$ be the Bergman space of the unit disc, and let $T_f$ denote the Bergman-space Toeplitz operator with analytic symbol $f$. For $f,g\in A^2(\mathbb{D})$, define the Berezin transform by

$$B f(z)=\int_{\mathbb{D}}\frac{f(\zeta)(1-|z|^2)^2}{|1-\overline{\zeta}z|^4}\,dA(\zeta).$$

Sarason's conjecture. The operator $T_fT_g^*$ is bounded on $A^2(\mathbb{D})$ if and only if

$$b_{f,g}:=\sup_{z\in\mathbb{D}}B(|f|^2)(z)B(|g|^2)(z)<\infty.$$

The condition is the natural two-weight analogue of the Bekollé–Bonami $B_2$ condition for the Bergman projection. The conjecture was disproved by F. Nazarov in 1997.