TITLE: Converse of a result of Koblitz and Ogus on algebraic products of gamma values
QUESTION [8 upvotes]: 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

REPLY [6 votes]: A convenient way to formulate this kind of questions is to use the language of distributions introduced by Kubert and Lang. I gave a short account in this answer to a previous MO question.
Your question is equivalent to asking whether the $\Gamma$ distribution is universal. Rohrlich has conjectured that this is the case, see Lang, Cyclotomic fields I and II, Chapter 2, Appendix, p. 66.
Lang has made the stronger conjecture that every polynomial relation between the values of $\Gamma$ at rational arguments is in the ideal generated by the standard relations, see $\S$3.1.5 in this survey of Michel Waldschmidt.
It seems that these conjectures are wide open. According to Waldschmidt (see $\S$5.3 in the same survey), already the case $n=5$ seems open.