χ^{cw} counts Tait colorings on arbitrary surfaces

Establish that for any surface Σ and trivalent graph Γ embedded in Σ, the trivial surrounding lattice TFT χ^{cw}: Bord_2^{def,cw}(𝔇_{+}^{3}) → Vect_F(ℂ) evaluates on (Σ, Γ) to the number of Tait colorings of Γ, i.e., prove χ^{cw}(Σ, Γ)(1) = #Tait(Γ).

Background

The main theorem of the paper proves this equality for planar trivalent graphs Γ ⊂ 𝕊^2. The authors conjecture the same equality for trivalent graphs embedded in any surface Σ, motivated by the expectation that edge-colorability is intrinsic to the graph and independent of the embedding surface.

A proof would extend the TFT-based counting result beyond the sphere and align χ^{cw} with the purely combinatorial Tait-coloring count on arbitrary surfaces.