Gaussian measures of dilations of convex rotationally symmetric sets in C^n

We consider the complex case of the so-called S-inequality. It concerns the behaviour of the Gaussian measures of dilations of convex and rotationally symmetric sets in C^n (rotational symmetry is invariance under the multiplication by $e^{it}$, for any real t). We pose and discuss a conjecture that among all such sets the measure of cylinders (i.e. the sets $\{z \in C^n: |z_1| \leq p\}$) decrease the fastest under dilations. Our main result of the paper is that this conjecture holds under the additional assumption that the Gaussian measure of considered sets is not greater than some constant c > 0.64.