Algebraic values of the continued-fraction set of multiple zeta-star values

Let $coverline{\mathbb{Q}}$ denote the set of algebraic numbers in $cmathbb{C}$, and let $ceta(\mathcal{D}_2)$ be the set appearing in the paper.

Algebraicity conjecture.

$$\eta(\mathcal{D}_2)\cap \overline{\mathbb{Q}}=\{n\mid n\in\mathbb{Z},\ n\geq 2\}.$$

This conjecture predicts that the only algebraic elements of $ceta(\mathcal{D}_2)$ are the integers at least $2$; it is presented as supported by extensive calculations, with no proof or resolution supplied here.