(In connection with this. My previous question was answered, and here is a modified version of my question.)
The Gamma function satisfies the relation $z\Gamma(z)=\Gamma(z+1)$, whence $|\Gamma(z+1)|>|\Gamma(z)|$ whenever $|z|>1$ (and $z$ is not a non-positive integer). Is the function $|\Gamma(x+iy)|$ of the variable $x$ increasing if the parameter $y$ is sufficiently large? I am also interested in a description of the area where (the local version of) this property is fulfilled.