Lang–Rohrlich conjecture for algebraic values of the Gamma function

Let $z\in\overline{\mathbb Q}$ be an algebraic number, and let $\Gamma$ denote the Gamma function.

Lang–Rohrlich conjecture. The value $\Gamma(z)$ is algebraic if and only if

$$z\in\mathbb Z_{\geq 1}.$$

Since $\Gamma(n)=(n-1)!$ for positive integers $n$, the conjecture says that the positive integers are exactly the algebraic arguments at which the Gamma function takes algebraic values. The source notes that the transcendence of $\Gamma(1/5)$ remains open, while the cases $\Gamma(1/3)$ and $\Gamma(1/4)$ are known to be transcendental.