The CTI–Schröder polynomial equality conjecture

Let $S_{n,d}$ be the complete split graph, let $\mathfrak{F}_{n,d}^{CTI}(q,t)$ be the $q,t$-CTI polynomial, and let $\mathsf{Schr\ddot{o}der}_{n,d}$ be the set of $(n,d)$-Schröder paths with statistics $\mathsf{area}$ and $\mathsf{bounce}^{\mathsf{Sch}}$. CTI–Schröder equality conjecture.

$$\mathfrak{F}_{n,d}^{CTI}(q,t)=\sum_{w\in\mathsf{Schr\ddot{o}der}_{n,d}}q^{\mathsf{area}(w)}t^{\mathsf{bounce}^{\mathsf{Sch}}(w)}.$$

This is equivalent to the CTI–ITC equality conjecture because the paper proves equality of the $q,t$-Schröder and $q,t$-ITC polynomials. The general equality remains open, although special cases are established.