Integral algebraic values in the sets of multiple zeta-star values

Let $coverline{\mathbb{Q}}$ denote the set of algebraic numbers in $cmathbb{C}$, let $ceta(\mathcal{D}_q)$ be the set associated with parameter $q$, and let $cmathbb{Z}$ denote the integers.

Integral algebraicity conjecture. For every integer $q\geq 2$,

$$\eta(\mathcal{D}_q)\cap(\overline{\mathbb{Q}}\setminus\mathbb{Z})=\varnothing.$$

This asserts that no nonintegral algebraic number belongs to any $ceta(\mathcal{D}_q)$ with $q\geq 2$. It is stated as a further conjectural expectation, and the supplied text gives no resolution.