Existence of compact sets with a prescribed hierarchy of vanishing ideals

Let $R(K)$ denote the uniform algebra of rational functions with poles off a compact set $K\subset\mathbb{C}$. For $x\in K$, write $M_x$ for the maximal ideal of functions in $R(K)$ vanishing at $x$, and $J_x$ for the ideal of functions vanishing on a neighborhood of $x$ in $K$. Hierarchy conjecture. For each integer $s\geq 2$, there exists a compact set $K\subset\mathbb{C}$ such that

$$\overline{J_x}\supset M_x^s$$

for every $x\in K$, but for some $y\in K$,

$$\overline{J_x}\not\supset M_y^{s-1}.$$

Consequently, for every $x\in K$ one has

$$\overline{J_x}=\overline{M_x^s}=\overline{M_x^{s+1}}=\overline{M_x^{s+2}}=\cdots,$$

and at the exceptional point $y$,

$$M_y\supsetneq\overline{M_y^2}\supsetneq\cdots\supsetneq\overline{M_y^s}=\overline{J_y}.$$

The conjecture would produce rational function algebras in which the powers of a maximal ideal have a strict closure hierarchy, separating the condition involving $\overline{J_x}\supset M_x^s$ from the corresponding condition involving squares of maximal ideals. The source notes that the converse implication is expected to fail and that an example for the opposite implication is planned for future work.