Interior empty-row and column insertion conjecture

Prove that, for every forbidden 0-1 matrix P and every matrix P′ obtained by inserting empty rows and columns strictly between the first and last rows and columns of P, sat(n, P′)=O(sat(n, P)) and sat(m0,n,P′)=O(sat(m0,n,P)).

Background

The paper proves preservation results for inserting empty columns in the fixed-dimension setting and for inserting empty rows under additional witness conditions. Conjecture 4.8 proposes that these restrictions are unnecessary and that inserting empty rows and columns in the interior changes the saturation functions by at most a constant-factor asymptotic bound.

References

We make the following conjecture, which would widely generalize the results of this section. Conjecture 4.8. Let P be any pattern, and let P ′ be derived from P by adding empty rows and columns in the interior. Then, sat(n, P ′) = O(sat(n, P )) and sat(m0, n, P ′) = O(sat(m0, n, P )).

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Conjecture 4.8, Section 4 (p. 13)