Biholomorphic Geometric Arveson-Douglas conjecture for quotient submodules

Let $V\subset\mathbb{B}^n$ be an affine homogeneous algebraic variety, meaning that there exist finitely many holomorphic homogeneous polynomials $q_1,\ldots,q_m$ such that

$$V=\{z\in\mathbb{B}^n:q_1(z)=\cdots=q_m(z)=0\}.$$

Let $(M_V^\bot,S_\phi)$ be the quotient submodule with compressed coordinate multipliers associated to $\phi\in\operatorname{Aut}(\mathbb{B}^n)$. Biholomorphic Geometric Arveson-Douglas conjecture. For some, equivalently every, $\phi\in\operatorname{Aut}(\mathbb{B}^n)$, the quotient submodule $(M_V^\bot,S_\phi)$ is $p$-essentially normal for all $p>\dim_{\mathbb{C}}V$. This formulation expresses biholomorphic invariance of the predicted essential-normality bound for affine homogeneous varieties; the source presents it as an equivalent formulation and does not establish it in general.