The conjecture relating cubic roots to ring class fields

Let $d$ be as in the paper, let $\mathcal{O}(d)$ be the associated quadratic order, and let $P_d$ and $R_d$ be the subsets defined there. For $\beta\in\mathcal{O}(d)$, let $c=c(\beta)$ be the product of the distinct rational primes dividing $\beta$ but not $d$, and let $M_c$ be the ring class field for the order $\mathbb{Z}[c\sqrt{-3d}]$. Assume that $\beta$ has a cubic norm and that $3\nmid N(\beta)$. Cubic-root class-field conjecture. One has

$$\beta^{1/3}\in M_c\quad\Longleftrightarrow\quad \beta\in P_d.$$

In particular, this equivalence holds for every $\beta\in R_d$ with $3\nmid N(\beta)$. This conjecture is introduced as the conditional input for the extension of the preceding theorem beyond the hypothesis $3\nmid h(d)$; numerical evidence is cited later in the paper.