Special cases of the Rohrlich-Lang conjecture with Gamma function The Gamma function satisfies:
$$\Gamma(z + 1) = z\Gamma(z)$$
$$\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}$$
$$\prod_{k = 0}^{n-1}\Gamma\left(a+\frac{k}{n}\right) = (2\pi)^{(n-1)/2}n^{-na + 1/2}\Gamma(na)$$

The Rohrlich conjecture states:
Any multiplicative dependence relation of the form
$$\pi^{b/2}\prod_{a \in \mathbb{Q}}\Gamma(a)^{m_a} \in \bar{\mathbb{Q}}$$
is a consequence of the relations above.

Do we know special cases where this is valid? For instance, given rationals $a$ and $b$, do we know when $\dfrac{1}{\pi}\dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a + b)}$ is algebraic? And for three rationals?
At least two of gamma(1/5), gamma(2/5) and pi are algebraically independent. ..
At least two of gamma(1/5), gamma(2/5) and pi are algebraically independent. ..