Sandpile avalanches in two dimensions are finite

In the abelian sandpile model on $\mathbb{Z}^d$, a site with at least $2d$ grains topples by sending one grain to each neighbor. In a box $[-L,L]^d$ whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain, and Athreya and Járai (2004) constructed the stationa