Subparking-function and acyclic-cut orientation conjecture for graph Tutte polynomials

Let $G$ be a graph, let $q\in V(G)$ be a sink, and let $T$ be an ordered, $q$-rooted spanning tree of $G$. For $U\subseteq V^{q}(G)$, write

$$\mathbf{x}^{U,T}:=\prod_{u\in U}x_u^{d^{G}_{U,T}(u)},\qquad d^{G}_{U,T}(u):=\left|\{e=\{u,v\}\in E(U,U^c):e\text{ is not the minimum edge in }E(U,U^c)\}\right|,$$

and let $I^{-1}_{(G,q,T)}=\langle \mathbf{x}^{U,T}:\emptyset\ne U\subseteq V^{q}(G)\rangle$. Let $\mathrm{PF}^{-}(G,q,T)$ be the set of $(G,q,T)$-subparking functions, and let $\mathrm{ACI}(G,q,T)$ be the set of acyclic-cut internal partial orientations of the rooted graph. Write $R$ for the polynomial ring in the variables $x_u$ indexed by $V^{q}(G)$, $g$ for the genus of $G$, and $D_{\mathcal O}$ for the divisor associated with an acyclic-cut internal partial orientation $\mathcal O$.

Subparking-function and acyclic-cut orientation conjecture. For any graph $G$ and choice of sink $q\in V(G)$, there exists an ordered, $q$-rooted spanning tree $T$ of $G$ such that

$$\operatorname{Hilb}(R/I^{-1}_{(G,q,T)};y)=y^{g}\cdot T_G(0,1/y)$$

and

$$\mathrm{PF}^{-}(G,q,T)=\{(D_{\mathcal O})_{\mathbb ZV^{q}(G)}:\mathcal O\in\mathrm{ACI}(G,q,T)\}.$$

This conjecture proposes a spanning-tree ordering simultaneously realizing the Tutte-polynomial Hilbert series and identifying subparking functions with divisors arising from acyclic-cut internal partial orientations. The parser supplies no evidence of resolution, so its status is open.