The totally sorted configuration and SYT probability conjecture for labeled chip-firing on star graphs

Let $\Delta^{k,m}$ be the star graph with $k$ branches, each receiving $m$ labeled chips, and consider the random labeled chip-firing process in which every firable vertex is equally likely to be selected at each stage and, after a vertex is selected, each chip there is equally likely to fire. A stable configuration is called totally sorted when chips $(1)$ through $(m)$ land on branch 1, chips $(m+1)$ through $(2m)$ land on branch 2, and so on, with chips $\left((k-1)m+1\right)$ through $(km)$ landing on branch $k$. Totally sorted configuration and SYT probability conjecture. The totally sorted configuration is the most likely stable configuration, and stable configurations corresponding to standard Young tableaux are more likely to occur than non-SYT configurations. This asks for a probabilistic refinement of the paper's structural results on reachable stable configurations: although reachable configurations can be mapped to row-increasing tableaux with increasing first and last columns, these conditions do not characterize all reachable configurations, and the conjectured probability comparison remains open.