Finite forbidden-subgraph characterization of the class $\mathfrak{D}_4$

Let $\mathfrak{D}_4$ be the graph class referenced in the paper, consisting of the relevant maximal triangle-free graphs satisfying property $\mathcal{D}_4$.

Finite forbidden-subgraph conjecture for $\mathfrak{D}_4$. There exists a finite family $\mathscr{F}$ of graphs such that $\mathfrak{D}_4$ is the class of maximal triangle-free graphs with at least two vertices that do not possess induced subgraphs in $\mathscr{F}$.

The paper records that several forbidden induced subgraphs are known for $\mathfrak{D}_4$, but that the analysis had not been completed. A finite characterization could provide an alternative proof of the paper's structural theorem.