Bounded semisaturation with witnesses but linear saturation

Determine whether there exists a forbidden 0-1 matrix with bounded semisaturation function and both vertical and horizontal witnesses, but with linear saturation function.

Background

The paper gives examples of matrices with linear saturation despite possessing both vertical and horizontal witnesses, but those examples also have linear semisaturation. It remains unknown whether bounded semisaturation together with both types of witnesses can coexist with linear saturation, which would clarify how far the hypotheses of Theorem 3.6 can be weakened.

References

This prompts the natural question of whether or not there exists a matrix that has a constant semisaturation function and vertical and horizontal witnesses, but linear saturation function. In other words, can the condition of strong indecomposability and the conditions on the witnesses in Theorem 3.6 be weakened to imply that the semisaturation function is bounded? This question also remains open.

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Section 10, Discussion (p. 39)