Finite generation conjecture for the module $\mathscr{M}_{\zeta}$

Let $S$ be the set of algebraic numbers under consideration, let $c13\in S$, and let $c13\mathscr{M}_{\zeta}$ be the associated set or module. Write $c13\mathbb{Z}[\zeta,\zeta^{-1}]$ for the Laurent polynomial ring generated by $c13\zeta$. Let $c13\Delta(P_{\zeta})$ denote the discriminant appearing in the definition of the fractional overring notation $c13\mathopen{\prec}\cdot\mathclose{\succ}$.

Finite generation conjecture. For every $c13\zeta\in S$, the set $c13\mathscr{M}_{\zeta}$ is finitely generated as a $c13\mathbb{Z}[\zeta,\zeta^{-1}]$-module. More generally,

$$\mathscr{M}_{\zeta}\subset \mathbb{Z}[\zeta,\zeta^{-1}]\mathopen{\prec}1/\Delta(P_{\zeta})\mathclose{\succ}.$$

The assertion concerns the proposed module structure of $c13\mathscr{M}_{\zeta}$; the surrounding discussion notes that the Laurent polynomial ring is Noetherian, so the displayed containment would imply finite generation. The supplied text does not establish the conjecture or provide evidence of its resolution.