Complete-bipartite spanning-subgraph conjecture for extremal flag Betti numbers

Let $\mathbf{G}(n,m)$ denote the collection of graphs on $n$ vertices and $m$ edges, and let $\beta^{\mathrm{FL}}_1(G)$ denote the first flag-complex Betti number of a graph $G$.

Complete-bipartite spanning-subgraph conjecture. If $G\in\mathbf{G}(n,m)$ and

$$\beta^{\mathrm{FL}}_1(G)=\max_{H\in\mathbf{G}(n,m)}\beta^{\mathrm{FL}}_1(H),$$

then $G$ contains a complete bipartite spanning subgraph.

This conjecture proposes a structural characterization of graphs attaining the extremal first flag-complex Betti number for fixed numbers of vertices and edges. The source presents the problem as a challenging direction for future research and gives an example showing that extremal graphs need not contain a Turán graph as a spanning subgraph.