Basis conjecture for the alternating component of generalized diagonal coinvariants

Let $m,n$ be positive integers, let $L_{mn,n}^+$ denote the set of rational $m$-Dyck paths, and let $L_{n,n}^+$ denote the set of Dyck paths of size $n$. For any rational Dyck path $\pi\in L_{mn,n}^+$ and any $i\in\{0,1,\dots,m-1\}$, let $\pi^i\in L_{n,n}^+$ be the Dyck path defined by the decomposition in the source, and let $\Delta_{\zeta^{-1}(\pi^i)}$ denote the associated alternating generalized diagonal coinvariant. Basis conjecture. The collection

$$\left\{\prod_{i=0}^{m-1}\Delta_{\zeta^{-1}(\pi^i)}\right\}$$

over all $m$-Dyck paths forms a basis of the alternating component of the space of generalized diagonal coinvariants. This conjecture proposes a combinatorial basis indexed by the decomposition of rational Dyck paths; its resolution depends on identifying the stated products with a basis of the alternating generalized diagonal coinvariant component.