Basis conjecture for the functions associated with two-step equivalence classes

For $k=0,1,2$, let $\{f^{(k)}\}$ be the set of functions realized as

$$\sum_{T\in\mathcal{C}}F_{\operatorname{ID}(T)},$$

where $\mathcal{C}$ is a single equivalence class of $\equiv^k$. Basis conjecture. The set $\{f^{(2)}\}$ forms a basis for the quasisymmetric functions. This concerns the generating functions associated with the equivalence classes introduced in the paper; the supplied text does not indicate whether the assertion has been proved or remains open.