The stable-structure tail conjecture for median dynamics on the square lattice

Consider median dynamics on $\mathbb{Z}^2$, with $\eta_\infty(0)$ denoting the limiting opinion at the origin. Stable-structure tail conjecture.

$$\limsup_{\alpha\to0}\frac{\mathbb{P}[\eta_\infty(0)\in[0,\alpha]]}{\alpha^4}<\infty.$$

The exponent $4$ comes from the smallest stable structures, namely $2\times2$ squares, and the conjecture asserts that these structures determine the order of the density near zero.