Existence and uniqueness conjecture for det S_r

Prove the conjecture that for every r ≥ 2 and every integer d ≥ 1 there exists a nontrivial determinant-like linear functional det_{S_r}: ⊗_{1≤i_1<⋯<i_r≤r d} V_d → k, satisfying the vanishing condition that det_{S_r}(⊗ v_{i_1,…,i_r}) = 0 whenever there exist 1 ≤ x_1 < ⋯ < x_{r+1} ≤ r d with v_{x_1,…,x_r} = v_{x_1,…,x_{r−1},x_{r+1}} = ⋯ = v_{x_2,…,x_{r+1}}, and that this functional is unique up to multiplication by a nonzero scalar.

Background

The conjecture encapsulates both existence and uniqueness (up to a scalar) of det S_r with its defining vanishing property. It is closely related to structural properties of the exterior-like algebra Λ^S_r and, if true, would unify the theory of these determinant-like invariants across all r and d.

While existence is known in several cases, nontriviality for r ≥ 4 and general uniqueness remain unresolved, making this conjecture a central open direction.