  We prove Weibel's conjecture for twisted $K$-theory when twisting by a smooth
proper connective dg-algebra. Our main contribution is showing we can kill a
negative twisted $K$-theory class using a projective birational morphism (in
the same twisted setting). We extend the vanishing result to relative twisted
$K$-theory of a smooth affine morphism and describe counter examples to some
similar extensions.
