Bloch–Srinivas conjecture on the stabilisation of relative $K$-theory filtrations

Let $X'\to X$ be a desingularisation of an integral variety $X$, with exceptional fibre $E\hookrightarrow X'$. For a positive integer $n$, let $F^dK_0(X',nE)$ denote the codimension-$d$ part of the $K$-theory of $X'$ relative to the thickening $nE$, and let $F^dK_0(X)$ be the corresponding filtration piece for $X$. Bloch–Srinivas conjecture. The inverse system

$$F^dK_0(X',E)\longleftarrow F^dK_0(X',2E)\longleftarrow F^dK_0(X',3E)\longleftarrow\cdots$$

stabilises, with stable value $F^dK_0(X)$. This conjecture concerns the relationship between zero-cycle-type information on a singular variety and the relative $K$-theory of a desingularisation along its exceptional fibre; the supplied text does not state whether it has been resolved.