Conjecture on algebraicity of nonmaximal-order Shintani--Faddeev RM values

Let $\beta\in\mathbb R$ satisfy $a\beta^2+b\beta+c=0$ with $a,b,c\in\mathbb Z$ and $b^2-4ac$ nonsquare, and let $\r\in\mathbb Q^2$. Let \shin^{\r}[\beta] and \samech^{\r}[\beta] denote the associated Shintani--Faddeev RM values. Let $\mathcal O$ be the order relevant to $(\r,\beta)$, let $\mm$ be an $\mathcal O$-invertible ideal, and suppose $(\r,\beta)\in\mathcal M_{\mathcal O,\mm}$. Algebraicity conjecture. Then \shin^{\r}[\beta] is an algebraic unit in an abelian extension of $\mathbb Q(\beta)$, and

\samech^{\r}[\beta]\in H_{\mm\infty_2}.

The claim removes the fundamental-discriminant hypothesis from the preceding theorem and extends its algebraicity and class-field containment to suitable nonmaximal orders; numerical evidence is cited, but the conjecture is not resolved.