Quadratic form theory over preordered von Neumann-regular rings

To each pair, R,T , consisting of a unitary commutative von Neumann-regular ring, R, where 2 is a unit and T is a preorder on R, we associate a reduced special group, GT(R), which faithfully reflects quadratic form theory, modulo T, over free R-modules and then show, using the representation of R as the ring of global sections of its affine scheme, together with results from [M. Dickmann, F. Miraglia, On quadratic forms whose total signature is zero mod 2n. Solution to a problem of M. Marshall, Invent. Math. 133 (1998) 243–278; M. Dickmann, F. Miraglia, Lamʹs Conjecture, Algebra Colloq. 10 (2003) 149–176; M. Dickmann, F. Miraglia, Algebraic K-theory of special groups, J. Pure Appl. Algebra 204 (2006) 195–234], that GT(R) satisfies a powerful K-theoretic property, the [SMC]-property. From this we conclude that quadratic form theory modulo T over free R-modules verifies Marshallʹs signature conjecture, Lamʹs conjecture, as well as a reduced version of Milnorʹs Witt ring conjecture.