The logarithmic tail conjecture for Kadanoff sandpile configurations

Let $D$ be a general parameter and let $N$ be any positive integer. Write $\pi(N)$ for the configuration obtained after $N$ grain additions, and let $\pi(N)_{[n,\infty[}$ denote its restriction to columns at least $n$. The notation $^*$ denotes Kleene closure, and $[0]$ denotes at most one zero.

Logarithmic tail conjecture. For a general parameter $D$ and all $N$, there exists a column $n$ in $O(\log N)$ such that

$$\pi(N)_{[n,\infty[}=(D-1,D-2,\dots,2,1)^*[0](D-1,D-2,\dots,2,1)^*0^\omega.$$

This conjecture generalizes the proved result for $D=3$, asserting that sufficiently far to the right, every configuration has a periodic tail built from the descending block $(D-1,D-2,\dots,1)$, with at most one exceptional zero, beginning at logarithmic distance from the origin. The paper reports that simulations suggest such a generalization, but leaves the claim open.