Treewidth 2 in the Planar Graph Product Structure Theorem

We prove that every planar graph is contained in $H_1⊠H_2⊠K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c\in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H)⩽2$, $G$ is not contained in $H⊠T⊠K_c$.