Strong Turán numbers of Cartesian products of paths

Determine whether the strong Turán number of the Cartesian product of paths satisfies \(\operatorname{st}(n,P_p\Box P_q)=\Theta(n^{3/2})\) for every pair of integers \(p\geq 3\) and \(q\geq 2\).

Background

The paper introduces the strong Turán number $\operatorname{st}(n,G)$, the maximum number of edges in an $n$-vertex graph containing no graph on $|G|$ vertices that strictly contains $G$. A central identity, $\operatorname{st}(n,G)=\operatorname{ex}(n,D(G))$, connects this parameter to classical Turán numbers, where $D(G)$ is the family of graphs obtained by adding one edge to $G$.

The authors propose estimating strong Turán numbers for Cartesian products of graphs as a general direction for further research. The concrete unresolved question concerns Cartesian products of two paths. They note that the analogous classical Turán estimate $\operatorname{ex}(n,P_t\Box P_t)=\Theta(n^{3/2})$ is known, while the proposed strong Turán estimate fails for $p=q=2$ because $P_2\Box P_2\cong C_4$ and $\operatorname{st}(n,C_4)=\Theta(n^2)$. They also state that the conjectured order is valid for the case $p=3,q=2$.