The full numerator criterion for pure stable polynomials

Let $p$ be a pure stable polynomial on $\mathbb{H}^2$, and let $p(0,0)=0$. Using the Puiseux factorization of $p$, write the associated product ideal as

$$\mathcal{I}=\prod_{j=1}^{k}(z_2+q_j(z_1),z_1^{2L_j})^{M_j}R_0.$$

Here $q_j$, $L_j$, $M_j$, and $R_0$ are the corresponding Puiseux-factorization data and local polynomial ring, and a function is locally $H^{\infty}$ when it is analytic and bounded on a neighborhood of $(0,0)$ intersected with $\mathbb{H}^2$.

Full numerator criterion. For every $f\in\mathbb{C}[z_1,z_2]$, $f/p$ is locally $H^{\infty}$ if and only if $f\in\mathcal{I}$.

The theorem preceding this conjecture proves the forward implication $f\in\mathcal{I}\Rightarrow f/p$ locally $H^{\infty}$ and the converse under special geometric hypotheses on $p$, including a double point, an ordinary multiple point, or repeated segments. The conjecture asserts the converse in general.