Uniqueness of det S2 beyond the known low-dimensional cases

Prove that for all integers d ≥ 5 the determinant-like map det_{S_2}: ⊗_{1≤i<j≤2d} V_d → k, characterized by vanishing when there exist indices 1 ≤ x < y < z ≤ 2d with v_{x,y} = v_{x,z} = v_{y,z}, is unique up to multiplication by a nonzero scalar.

Background

For r = 2, uniqueness up to a scalar has been established for d = 2, 3, and 4. The general case for larger d remains unresolved.

Settling uniqueness for det S2 in higher dimensions would complete the understanding of the 2-equilibrium determinant-like invariant and align it with the classical uniqueness property of the usual determinant.