Generalized Gersten injectivity conjecture for abstract blow-up squares

Let $A$ be a Noetherian local $\mathbb Q$-algebra, let $I\subseteq A$ be an ideal, let $n\in\mathbb Z$, and consider an abstract blow-up square

$$\begin{array}{ccc} Y &\longrightarrow& X\\ \downarrow && \downarrow\\ \operatorname{Spec}(A/I)&\longrightarrow&\operatorname{Spec}(A). \end{array}$$

Assume that $X$ is regular. Here $K_n(A,I)$, $K_n(A/I^r,I/I^r)$, and $K_n(X,Y_{\mathrm{red}})$ denote the corresponding relative algebraic $K$-groups. Generalized Gersten injectivity conjecture. The canonical map

$$K_n(A,I)\longrightarrow K_n(A/I^r,I/I^r)\oplus K_n(X,Y_{\mathrm{red}})$$

is injective for $r\gg 0$. This conjecture incorporates both infinitesimal thickenings and a resolution-like regular scheme in detecting the $K$-theory of a singular ring. The paper formulates such analogues and proves them in various cases, but the general assertion is not resolved in the supplied text.