Saturation under Kronecker products

Prove that if forbidden 0-1 matrices P and Q satisfy sat(n,P)=O(1) and ssat(n,Q)=O(1), then sat(n,P⊗Q)=O(1).

Background

The paper proves an exact boundedness characterization for semisaturation under Kronecker products, but shows that the analogous saturation statement does not hold in general in the forward direction. The reverse implication remains unresolved, motivating Conjecture 5.4.

References

However, we fail to resolve the reverse implication of Theorem 5.1 when analogous statements are made for saturation. Based on the known case for permutation matrices, we formulate the following conjecture. Conjecture 5.4. If P and Q are patterns with sat(n, P ) = O(1) and ssat(n, Q) = O(1), then sat(n, P ⊗ Q) = O(1).

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Conjecture 5.4, Section 5 (p. 15)