Here is a proof that $12$ is the smallest possible number of regions in any feasible solution.
Consider an arbitrary division of an arbitrary rectangle into $n$ regions, such that every region has exactly five neighboring regions. We translate this picture into a so-called planar graph: each of the $n$ regions then translates into a vertex/point, and there is a connecting edge between any two vertices/points that correspond to neighboring regions. The total number of edges is denoted $e$.
We only need facts on $e$ and $n$
Fact 1: $~~~~2e=5n$
This can be seen by counting the touching vertex-edge pairs: every edge touches two vertices, and every vertex touches $5$ edges.
Fact 2: $~~~~e\le3n-6$
This fact can be found (for instance) on the wiki-page in the paragraph on maximal planar graphs. If follows from the Euler formula for planar graphs.
Now by combining Facts 1 and 2, we get
which implies $0\le n/2-6$ and hence $n\ge12$. Consequently, Lopsy's construction is best possible with respect to the number of used regions.