Time Averages for the Vortex Model and Stroboscopic Ergodic Averages

We consider the vortex model on the plane, focusing on the case of vortices with the same sign and, for simplicity, assuming all vortices possess equal circulation. In particular we are interested at the time average of the vorticity density, i.e. the empirical measure associated to the vortices.
We conjecture that, for a.e. initial data, the time average of the empirical density is radial.
We prove the result for N=3 vortices by exploiting the integrability of the system.
For N > 3 vortices we motivate the conjecture by transforming the problem into the independence of ergodic stroboscopic averages from initial data along a single trajectory, when using a suitable rotation angle as the independent variable instead of the time variable.