The weight-two motivic complex of equicharacteristic local fields

Let L/K be a finite Galois extension of local fields in positive characteristic with group G. The weight-two motivic complex Γ(2, L) defines an element of ExtZ[G]2(K2(L),K3(L).). We show, after inverting the prime 2, that cup-product with this 2-extension induces an isomorphism on Tate cohomology. In fact we show that this isomorphism coincides with cup-product by the K2/K3 local fundamental class previously constructed.