The $(1,2)$ combinatorial Frobenius-series conjecture

Let $B_n^{(1,2)}$ be the proposed monomial basis of $R_n^{(1,2)}$. For $b\in B_n^{(1,2)}$, write $\deg_x(b)$, $\deg_\theta(b)$, and $\deg_\xi(b)$ for its degrees, and let $\operatorname{Asc}(b)$ be its ascent set. Let $Q_{S,n}$ denote the fundamental quasisymmetric function indexed by $S\subseteq\{1,\ldots,n-1\}$. Frobenius-series conjecture.

$$\operatorname{Frob}(R_n^{(1,2)};q;u,v)=\sum_{b\in B_n^{(1,2)}}u^{\deg_\theta(b)}v^{\deg_\xi(b)}q^{\deg_x(b)}Q_{\operatorname{Asc}(b),n}.$$

The formula is proposed in terms of the conjectural basis and ascent statistics; its validity remains open in the supplied text.