Transcendence conjecture for the Stirling-Ramanujan constants

Let $S_n$, $n\geq 0$, denote the Stirling-Ramanujan constants, with $S_0$ the Stirling constant, and let $\gamma=S_{-1}$ be the Euler–Mascheroni constant. The Glaisher–Kinkelin constant is $A=e^{S_1}$.

Stirling-Ramanujan transcendence conjecture. The Stirling-Ramanujan constants are transcendental.

The arithmetic nature of these constants is unknown; the conjecture encompasses the classical Euler–Mascheroni, Stirling, and Glaisher–Kinkelin constants as well as the higher constants.