Uniqueness of det S3 for d ≥ 3

Prove that for every integer d ≥ 3 the determinant-like map det_{S_3}: ⊗_{1≤i<j<k≤3d} V_d → k, defined by the property that det_{S_3}(⊗ v_{i,j,k}) = 0 whenever there exist 1 ≤ x < y < z < t ≤ 3d with v_{x,y,z} = v_{x,y,t} = v_{x,z,t} = v_{y,z,t}, is unique up to multiplication by a nonzero scalar.

Background

For r = 3, the map det S3 is known to be nontrivial and unique up to a scalar when d = 2. However, the uniqueness question for higher dimensions remains unresolved.

Resolving uniqueness for d ≥ 3 would complete the classification of det S3 across all dimensions and clarify the structure of the associated algebraic invariants.