Maximal total persistence conjecture for extremal flag-complex filtrations

Let an edgewise filtration of a flag complex be a filtration obtained by adding one edge at a time, and let total persistence in homology degree $1$ be the sum of the lengths of the intervals in its degree-$1$ persistence barcode. Let the extremal filtrations described in Section 4 be the filtrations referred to in the source.

Total-persistence conjecture. The extremal filtrations described in Section 4 achieve the maximal total persistence over any edgewise filtration of flag complexes in homology degree $1$.

The conjecture concerns the global optimization of total persistence rather than optimization of the Betti number at every filtration stage. The source notes that no filtration can maximize the first Betti number at all stages, so proving this conjecture likely requires new ideas; its resolution is not given.