Polynomial-time computation of the well-covered weight space for graphs with no 6- or 7-cycles

Let $G$ be a graph in $\mathcal{G}(\widehat{C}_{6},\widehat{C}_{7})$. The notation $WCW(G)$ denotes the vector space of all weight functions $w$ such that $G$ is $w$-well-covered.

Polynomial-time computation conjecture. The following problem can be solved in polynomial time: given a graph $G \in \mathcal{G}(\widehat{C}_{6},\widehat{C}_{7})$, output $WCW(G)$.

The authors motivate this conjecture by noting that enumerating all induced complete bipartite subgraphs can require exponential time. The paper proves polynomial-time recognition of generating subgraphs in this graph class, but leaves computation of the full well-covered weight space open.