The random labeled chip-firing probability conjecture

For each of the three random labeled chip-firing protocols described in the source—uniformly choosing a legal firing move, uniformly choosing an unstable vertex and then a pair of chips there, or uniformly choosing a labeled stabilization sequence—consider the probability that the initial configuration $\Delta^{2m+1}$ sorts. Random labeled chip-firing probability conjecture. With respect to any of these three protocols, this probability converges to

$$\frac{1}{3}$$

as $m\to\infty$. The conjecture predicts a common asymptotic sorting probability for three distinct randomization procedures. The source reports only limited simulations and gives no proof; it notes that the probability cannot have a limiting value greater than $1/3$.