Sarason's conjecture for the Hardy space

Let $H^2(\mathbb{T})$ be the Hardy space, and let $\mathcal{T}_f$ denote the Hardy-space Toeplitz operator with symbol $f$. For $f,g\in H^2(\mathbb{T})$, let $\mathcal{P}$ denote the Poisson extension of a function on $\mathbb{T}$ to $\mathbb{D}$.

Sarason's conjecture. The operator $\mathcal{T}_f\mathcal{T}_g^*$ is bounded on $H^2(\mathbb{T})$ if and only if

$$\sup_{z\in\mathbb{D}}\mathcal{P}(|f|^2)(z)\mathcal{P}(|g|^2)(z)<\infty.$$

This is the Hardy-space analogue of the Bergman-space Sarason conjecture and is related to the two-weight problem for the Riesz projection. F. Nazarov disproved it in 1997.