A bramble in a graph $G$ is a set of connected subgraphs $H_1, \dots, H_m$ such that for every $i, j$, either $H_i$ intersects $H_j$ in a vertex, or there exists an edge of $G$ with one end in $V(H_i)$ and one end in $V(H_j)$. The order of the bramble is the minimum $|X|$ such that $X \subseteq V(G)$ and $X \cap V(H_i) \neq \emptyset$ for all $i$. The bramble number of $G$ is the maximum order of a bramble in $G$. Denote it $Br(G)$.
Say a strict bramble in a graph $G$ is a set of connected subgraphs $H_1, \dots, H_m$ such that for all $i, j$, $H_i$ intersects $H_j$ in a vertex. Definite analogously the order of a strict bramble and the strict bramble number of a graph, and denote the strict bramble number by $sBr(G)$.
Clearly, the bramble number is at least the strict bramble number. They also won't be equal in general. For example, in $K_t$, the bramble number is $t$, but the strict bramble number will be essentially $t/2$.
Is this tight? Is it true that $sBr(G) \ge Br(G)/2$? The grid theorem implies that there is a function $f$ such that the $sBr(G) \ge f(Br(G))$, but it might be true as well with a linear function.