Nonzero χ^{cw} for bridgeless planar trivalent graphs

Prove that for every planar trivalent graph Γ without a bridge, the trivial surrounding lattice topological field theory χ^{cw}: Bord_2^{def,cw}(𝔇_{+}^{3}) → Vect_F(ℂ) evaluates on the closed surface with defects (𝕊^2, Γ) to a nonzero scalar, i.e., establish χ^{cw}(𝕊^2, Γ)(1) ≠ 0.

Background

The paper constructs a 2D defect lattice TFT χ^{cw} whose evaluation on a planar trivalent graph embedded in the sphere equals the number of Tait colorings (3-edge-colorings) when nonzero. For graphs with a bridge, χ^{cw}(𝕊^2, Γ)(1) = 0, and the authors conjecture the converse direction, which is equivalent to the four-color theorem via Tait’s correspondence.

This conjecture asks for a TFT-formulation of the four-color theorem: for any bridgeless planar cubic graph Γ, the partition function χ^{cw} on (𝕊^2, Γ) should be nonvanishing.