Let $n$ be a positive integer, $a_1,\ldots,a_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$, $$ \tag{$\star$} \sum_{i=1}^{n-1} i a_{(ui\!\!\!\mod n)} = 0. $$ Then the product of values of the Gamma function $$ \tag{$\star\star$} \prod_{i=1}^{n-1}\Gamma\left(\frac{i}{n}\right)^{a_i} $$ is known to be an algebraic number. This is a consequence of the reflection and multiplication formulas for $\Gamma$, and the result is proved in an appendix by Koblitz and Ogus to [1].
I would like to know if the converse of this result is expected to be true. Since the converse is almost certainly open if it is true, I will ask:
Are there any examples where the product $(\star\star)$ is known to be algebraic but $(\star)$ does not hold?
[1] Pierre Deligne, Valeurs de fonctions $L$ et périodes d'intégrales, Proceedings of Symposia in Pure Mathematics, Vol. 33 (1989), part 2, pp. 313-346
