The limit point in the Jante's law process has an absolutely continuous distribution

We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante's law process. We consider a version of the model where the space of possible opinions is a convex body $\mathcal{B}$ in $\mathbb{R}^d$. $N$ individuals in a population each hold a (multidimensional) opinion in $\mathcal{B}$. Repeatedly, the individual whose opinion is furthest from the center of mass of the $N$ current opinions chooses a new opinion, sampled uniformly at random from $\mathcal{B}$. Kennerberg and Volkov showed that the set of opinions that are not furthest from the center of mass converges to a random limit point. We show that the distribution of the limit opinion is continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.