  Suppose $\alpha$ is an orientation-preserving diffeomorphism (shift) of
$\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. In
\cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of
the singular integral operator with shift \[
(aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on $L^p(\mR_+)$ with
$1<p<\infty$, where $P_\pm=(I\pm S)/2$, $S$ is the Cauchy singular integral
operator, and $W_\alpha f=f\circ\alpha$ is the shift operator, under the
assumptions that the coefficients $a,b,c,d$ and the derivative $\alpha'$ of the
shift are bounded and continuous on $\mR_+$ and may admit discontinuities of
slowly oscillating type at $0$ and $\infty$. Now we prove that those conditions
are also necessary.
