Analogues of Gersten's conjecture for singular schemes

We formulate analogues, for Noetherian local $\mathbb{Q}$-algebras which are not necessarily regular, of the injectivity part of Gersten’s conjecture in algebraic $K$-theory, and prove them in various cases. Our results suggest that the algebraic $K$-theory of such a ring should be detected by combining the algebraic $K$-theory of both its regular locus and the infinitesimal thickenings of its singular locus.