The CTI–ITC polynomial equality conjecture for complete split graphs

Let $S_{n,d}$ be the complete split graph, and let $\mathfrak{F}_{n,d}^{CTI}(q,t)$ and $\mathfrak{F}_{n,d}^{ITC}(q,t)$ denote the $q,t$-CTI and $q,t$-ITC polynomials, respectively. CTI–ITC equality conjecture.

$$\mathfrak{F}_{n,d}^{CTI}(q,t)=\mathfrak{F}_{n,d}^{ITC}(q,t).$$

Equality is known in some special cases, but the general case remains unresolved. It is equivalent, using the established equality of the $q,t$-Schröder and $q,t$-ITC polynomials, to the Schröder-polynomial formulation appearing below.