Dimension filtrations in birational localisation

Let $S_b$ be the class of birational morphisms between smooth varieties over a field $F$, and let $L_n=S_b^{-1}d_{\leq n}\Sm(F)$. Kahn and Sujatha asked whether the natural functor $L_n\to S_b^{-1}\Sm(F)$ is fully faithful. We prove that it is fully faithful exactly for $n=0$. More strongly, for every $n\geq1$ and every $N\geq n+1$, the transition functor $L_n\to L_N$ has an infinite fibre on an endomorphism set. The proof identifies a sharp dimension threshold: if $\dim X+r\leq n$, then $X\times\mathbb A^r\to X$ is invertible in $L_n$ precisely when $\dim X+r\leq n-1$. We also give proper and projective analogues.