A statistic-preserving CTI–ITC bijection conjecture for sorted recurrent configurations

Let $S_{n,d}$ be the complete split graph, and let $\mathsf{SortedRec}(S_{n,d})$ be the set of sorted recurrent configurations on $S_{n,d}$. For a configuration $c$, write $\mathsf{height}(c)$ for its height, and write $\mathsf{wtopple}_{CTI}(c)$ and $\mathsf{wtopple}_{ITC}(c)$ for the CTI and ITC weighted-toppling statistics. Statistic-preserving bijection conjecture. There exists a bijection

$$\Psi:\mathsf{SortedRec}(S_{n,d})\mapsto\mathsf{SortedRec}(S_{n,d})$$

such that, for every $c\in\mathsf{SortedRec}(S_{n,d})$,

$$(\mathsf{height}(c),\mathsf{wtopple}_{CTI}(c))=(\mathsf{height}(\Psi(c)),\mathsf{wtopple}_{ITC}(\Psi(c))).$$

Such a bijection would explain the correspondence between the CTI and ITC polynomials directly in the sandpile model; the paper states that the authors expect such an explanation, but no bijection is currently known.