Radial time-average conjecture for the equal-vortex model

Let $z=(z_1,\ldots,z_N)$ be a configuration of $N$ equal vortices, each of vorticity $1$, and define the pre-shape set

$$\Gamma_N=\left\{z:\sum_{k=1}^N z_i=0,\,\frac{1}{N}\sum_{k=1}^N|z_i|^2=1\right\}.$$

For almost every $z\in\Gamma_N$, the time average of the empirical vorticity density exists in the weak-measure sense,

$$\mu(dz)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T\frac{1}{N}\sum_{i=1}^N\delta_{z-z_i(t)}\,dt,$$

and is radial, meaning invariant under rotations. Equivalently, for every smooth observable $\Phi:\mathbb{R}^2\rightarrow\mathbb{R}$ and every $\theta\in[0,2\pi)$, the time average of

$$\Psi_\theta(z)=\frac{1}{N}\sum_{k=1}^N\Phi\circ R_\theta(z_k)$$

exists and is independent of $\theta$. The conjecture concerns the long-time distribution of vortices on the fixed center-of-mass and inertial-moment level set; establishing existence and rotational invariance of these averages remains open.