Smallest odd cycle forced by planar two-colorings

Determine the smallest integer $\ell$ with $2\leq \ell\leq 13$ such that every 2-coloring of the Euclidean plane contains a monochromatic unit-copy of $C_{2\ell+1}$, and determine whether $\ell=2$.

Background

The paper proves that sufficiently long odd cycles are forced in planar 2-colorings and gives the explicit bound $\ell\geq 13$, while noting that the triangle is avoidable and the case of C_5 is unresolved.