The moduli-space conjecture for enriched curves

Let $Y$ be a nodal curve with $b4$ nodes, let $\Gamma$ be its dual graph, and let $\mathcal B$ be the set of bonds of $\Gamma$. Define the vector-weight collection \widetilde\mathcal A_Y=(\underline{a}_1,\ldots,\underline{a}_{\delta},\underline{b}_0,\underline{b}_{\infty}) by

$$a_{i,B}=\begin{cases}0&\text{if }e_i\notin B,\\ \frac{1}{|B|}&\text{if }e_i\in B,\end{cases}$$

for $\underline{a}_i=(a_{i,B})_{B\in\mathcal B}$, together with $\underline{b}_0=(1,1,\ldots,1)$ and $\underline{b}_{\infty}=(b_{i,B})_{B\in\mathcal B}$ where $b_{i,B}=1/|B|$. Let $\mathcal E_Y$ be the associated moduli space of enriched structures on $Y$, and let \overline{\mathcal M}_{0,\widetilde\mathcal A_Y} denote the moduli space of genus-zero curves with the indicated vector weights. The moduli-space conjecture for enriched curves. $\mathcal E_Y$ is isomorphic to \overline{\mathcal M}_{0,\widetilde\mathcal A_Y}.

This would describe the enriched-structure space as a generalized Hassett moduli space. The construction is motivated by examples in which $\mathcal E_Y$ is a Hassett moduli space, although the authors also note examples where it is not an ordinary Hassett space; the existence and identification of the generalized moduli space remain conjectural.