Eoh–Choi's corona-free characterization conjecture for graph grabbing

Let $G$ be a graph with a weight function $w\colon V(G)\to\mathbb{R}_{\geq 0}$ whose codomain is $\{0,1\}$. Let $\mathcal{C}_\text{odd}$ be the family of corona products of an odd cycle and a point. A graph is $\mathcal{C}_\text{odd}$-free if it has no induced subgraph isomorphic to a member of $\mathcal{C}_\text{odd}$. Eoh–Choi's conjecture. Alice wins the graph-grabbing game if and only if $G$ is $\mathcal{C}_\text{odd}$-free. Eoh and Choi proved the forward implication under the stated $\{0,1\}$-valued weighting setting; the conjecture asserts the converse as well and remains open.