Polygonal limiting-shape conjecture for the splitting automaton

Let the splitting automaton evolve on $\mathbb Z^d$ with parameter $h$, and let $n$ be its size parameter. A limiting shape is the deterministic rescaled asymptotic shape of the occupied set, as defined in the paper.

Polygonal limiting-shape conjecture. For every

$$\nh\in\left[1-\frac{3}{4d+2},1\right),$$

there exists $n_0$ such that, for every $n>n_0$, the limiting shape is a polygon depending only on $h$ and $d$.

The conjecture is motivated by simulations showing many possible shapes as $h$ varies, together with cases in which changing $n$ affects the observed behavior. The existence and asserted parameter-independence of the limiting polygon remain open in the stated range.