Rohrlich's conjecture on algebraic products of gamma values

Let $a_1, a_2, \ldots, a_r$ be rational numbers that are not in $\{0, -1, -2, \ldots\}$. Let $D$ be a common denominator of the $a_i$'s. Rohrlich's conjecture. The product

$$\Gamma(a_1) \Gamma(a_2) \cdots \Gamma(a_r)$$

is an algebraic multiple of $\pi^{r/2}$ if and only if, for all $m \in \{1, 2, \ldots, D-1\}$ relatively prime to $D$,

$$\sum_{1 \leq i \leq r} \{ma_i\} = r/2,$$

where $\{x\}$ denotes the fractional part of the real $x$. The conjecture predicts that algebraic relations among products and quotients of normalized gamma values derive from the functional equation, reflection formula, and multiplication formula for the gamma function; its general status is not established in the source.