Barcode-interval bound in arbitrary homological degree

Prove that for every filtration of flag complexes on n vertices, the number of intervals in its degree-k persistence barcode is at most the k-th Betti number of the flag complex of the Turán graph \(\mathcal{T}_{n,k+1}\).

Background

The paper establishes sharp extremal results for individual Betti numbers and, in homological degree one, for the maximal number of barcode intervals. It does not establish the analogous interval-count bound in arbitrary homological degree k. The proposed inequality would extend the degree-one result by relating the number of persistent homology intervals to the extremal Betti number realized by the Turán graph $\mathcal{T}_{n,k+1}$.