The predicted equality between ray class group exponents and unit quotient orders

Let $\mathfrak{a}\subseteq\mathfrak{o}$ be a $G$-stable ideal. Write $E_{\mathfrak{a}}$ for the relevant ray-unit group, $\mathcal{C}_{\mathfrak{a}}=\{\delta\in\mathcal{C}(\mathfrak{a}):\delta\equiv1\bmod{\mathfrak{a}}\}$, and let $\mathfrak{C}_{\mathfrak{a}}$ denote the associated ray class group. For a prime $p$ and character $\varrho$, write $\operatorname{Syl}_p(-)_{\varrho}$ for the $\varrho$-component of the Sylow $p$-subgroup and $\operatorname{exp}(-)$ for its exponent. The predicted equality. If $p\nmid2\cdot|G|$, then

$$\left|\operatorname{Syl}_p(E_{\mathfrak{a}}/\mathcal{C}_{\mathfrak{a}})_{\varrho}\right|=\operatorname{exp}\left(\operatorname{Syl}_p(\mathfrak{C}_{\mathfrak{a}})_{\varrho}\right).$$

This is presented as a possible stronger consequence of the article's annihilation results. The statement concerns the relationship between ray-unit quotients and ray class groups; its resolution is not given in the source.