Albertson–Stromquist conjecture for chromatic numbers of 2-manifolds

Prove that every 2-manifold has chromatic number at most 5, thereby resolving the conjecture attributed to Albertson and Stromquist.

Background

The paper discusses chromatic numbers of softly Barycentrically refined manifolds and notes that 2-manifolds can have chromatic number larger than the dimension-based lower bound. The cited Albertson–Stromquist conjecture asserts the universal upper bound c(G) ≤ 5 for 2-manifolds. The conjecture is presented as unresolved in the paper and is not proved by the results establishing the bounds for soft Barycentric refinements.