Poisson attraction conjecture for one-dimensional coarsening

Let $\nu$ be a probability measure on $\mathbb R$ with finite mean $m<\infty$, and let $Q_\nu$ be the stationary renewal process on $\mathbb R$ with renewal distribution $\nu$. A weak solution with initial conditions $Q_\nu$ is a stochastic evolution satisfying the paper's evolution rule, and let $\nu_t$ denote its time-$t$ law after rescaling space by $e^{-2t}$. Poisson attraction conjecture. For every such $\nu$, there exists a weak solution ${\mathbb P}_\nu$ with initial conditions $Q_\nu$, and, for its rescaled time-$t$ law,

$$\nu_t\mathrel{\longrightarrow}{\mathcal P}(1/m)\quad\text{weakly as }t\to\infty.$$

Here ${\mathcal P}(1/m)$ denotes the Poisson law of intensity $1/m$. This conjecture formalizes the numerical observation that the Poisson distribution is an attracting fixed point for the rescaled dynamics, extending the invariant Poisson case to arbitrary renewal initial data.