Is Euler-Mascheroni Constant Irrational?

Open Question
It is not even known whether the Euler-Mascheroni constant $\gamma$ is irrational, let alone whether it is transcendental or not.

Progress
In $1977$ it was established by that if $\gamma$ is a rational number expressible as the ratio of two integers $\dfrac a b$, it would be necessary for $b$ to be greater than $10^{10 \, 000}$.

By $1980$ that lower limit had been raised, according to in collaboration with, to $10^{15 \, 000}$.