The D-multidissection basis conjecture for $V^D(n)$

For $n\geq 2$, label the vertices of the centrally symmetric polygon ${\Bbb P}_{2n}$ clockwise by $1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}$. Let $V^D(n)$ be the quotient space defined in the paper, and for each D-multidissection $f$ of ${\Bbb P}_{2n}$ let

$$z_f^D:=\prod_{e\in E_D}(z_e^D)^{f(e)},$$

where the elements $z_e^D\in V^D(n)$ are defined from the D-edges as in the preceding construction. D-multidissection basis conjecture. The set

$$\{z_f^D\},$$

where $f$ ranges over all D-multidissections of ${\Bbb P}_{2n}$, is a ${\Bbb C}$-basis for $V^D(n)$. This conjecture would provide a purely algebraic route to the proof of the preceding theorem, replacing the separate algebraic and combinatorial ingredients used there.