Exact saturation for triangular matrices

Prove or determine the exact saturation function for triangular k×k 0-1 matrices with all diagonal entries equal to one, including the conjectured formula sat(m,n,T)=2m+2n−4 for k=3 and the conditions guaranteeing sat(m,n,T)=(k−1)(m+n−k+1) for k>3 and m,n≥k.

Background

A binary linear-programming model produces small values of the saturation function and motivates conjectured formulas for triangular patterns with full diagonal. The authors explicitly state a conjecture for 3×3 triangular matrices and ask what additional conditions ensure the general upper bound is sharp for larger triangular matrices.

References

Problem 8.2. Let T be any k × k triangular 0-1 matrix with all ones on the diagonal. If k = 3, we conjecture that sat(m, n, T ) = 2m + 2n − 4 for m, n ≥ 3. For k > 3, what conditions need to be placed to ensure that sat(m, n, T ) = (k−1)(m+n−k+1) for m, n ≥ k?

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Problem 8.2, Section 8 (p. 27)