Global maximality of the degree-one extremal filtrations

Prove that the extremal degree-one edgewise filtrations of the complete graph \(K_n\) described in Section 5 achieve maximal total persistence among all edgewise filtrations of flag complexes in homology degree one.

Background

The paper characterizes edgewise filtrations of $K_n$ that simultaneously have a maximal number of degree-one barcode intervals and maximal total persistence among filtrations containing the Turán graph $\mathcal{T}_{n,2}$. The unresolved issue is whether these same filtrations remain globally optimal when compared with every edgewise filtration of flag complexes, including filtrations whose intermediate graphs do not contain $\mathcal{T}_{n,2}$ as a spanning subgraph. The authors note that a fiberwise-optimal filtration cannot maximize the first Betti number at every filtration step, which makes the total-persistence claim nontrivial.