Conjecture on the transcendence of Euler's constant

Define Euler's constant by

$$\gamma=\lim_{n\to\infty}\left(1+\frac12+\frac13+\cdots+\frac1n-\log n\right).$$

Transcendence conjecture for Euler's constant. The number $\gamma$ is transcendental. The source notes that even irrationality of $\gamma$ is not known, so this stronger conjecture remains open.