R. Yang's Hilbert-Schmidt conjecture for finitely generated submodules

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and let $M$ be a submodule. Define

$$\Sigma_0^M:=\|[R_z^*,R_z][R_w^*,R_w]\|_{H.S.}^2,\qquad \Sigma_1^M:=\|[R_z^*,R_w]\|_{H.S.}^2,$$

where $R_z$ and $R_w$ are multiplication by the coordinate functions on $M$. The submodule $M$ is Hilbert-Schmidt when $\Sigma_0^M$ and $\Sigma_1^M$ are finite. R. Yang's Hilbert-Schmidt conjecture. Every finitely generated submodule $M$ of $H^2(\mathbb{D}^2)$ is Hilbert-Schmidt. This was known for finitely generated polynomial submodules and for submodules satisfying certain additional conditions; the conjecture remains open in general.