Cyclicity threshold conjecture for polynomials in Dirichlet-type spaces of the unit ball

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, let $\mathbb S_n$ be its boundary sphere, and let $\mathcal{Z}(p)$ denote the zero set of a polynomial $p$. A polynomial is cyclic in $D_\alpha(\mathbb B_n)$ if the polynomials obtained by multiplying it by elements of the polynomial algebra are dense in $D_\alpha(\mathbb B_n)$. Let $p\in\mathbb C[z_1,\dots,z_n]$ be a polynomial with no zeros in $\mathbb B_n$. Suppose that $\mathcal{Z}(p)\cap\mathbb S_n$ contains a real submanifold of $\mathbb R^{2n}$ of dimension $m-1$, where $m=2,3,\dots,n$, but contains no submanifold of any higher dimension. Cyclicity conjecture. Then $p$ is cyclic in $D_\alpha(\mathbb B_n)$ if and only if

$$\alpha\le\frac{2n-(m-1)}{2}.$$

The conjecture extends the known characterization for the model polynomials $1-m^{m/2}z_1\cdots z_m$, whose boundary zero sets have real dimension $m-1$. It predicts that the maximal real dimension of the boundary zero set determines the precise cyclicity threshold for arbitrary zero-free polynomials, while the claim remains unresolved in the stated generality.