Seymour's projective-cube homomorphism conjecture for planar graphs

Let $G$ be a planar graph and let $k$ be a positive integer. Write $PQ_{2k}$ for the $(2k)$-dimensional projective cube, obtained from the $(2k+1)$-dimensional cube by identifying antipodal vertices. A graph homomorphism $G\to PQ_{2k}$ is a vertex map preserving edges.

Seymour's conjecture. Every planar graph whose odd cycles all have length at least $2k+1$ has a homomorphism to $PQ_{2k}$.

The case $k=1$ is equivalent to the Four Color Theorem, since $PQ_2\cong K_4$. The case $k=2$, concerning the Clebsch graph $PQ_4$, was resolved affirmatively by Naserasr using a theorem of Guenin; the general conjecture is therefore solved in the database sense.