R. Yang's uniform invariant conjecture for rank-one submodules

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and for a submodule $M$ 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 $M$. Let $\mathcal{S}$ be the set of all rank-one submodules of $H^2(\mathbb{D}^2)$. R. Yang's uniform invariant conjecture.

$$\sup_{M\in\mathcal{S}}\left(\Sigma_0^M+\Sigma_1^M\right)<\infty.$$

This asks for a uniform bound on the Hilbert-Schmidt invariants over all rank-one submodules. The source presents it as a more general conjecture, and no resolution is supplied here.