  Suppose $\Gamma$ is a Carleson Jordan curve with logarithmic whirl points,
  $\varrho$ is a Khvedelidze weight, $p:\Gamma\to(1,\infty)$ is a continuous
function satisfying $|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|$ for
$|\tau-t|\le 1/2$, and $L^{p(\cdot)}(\Gamma,\varrho)$ is a weighted generalized
Lebesgue space with variable exponent. We prove that all semi-Fredholm
operators in the algebra of singular integral operators with $N\times N$ matrix
piecewise continuous coefficients are Fredholm on
$L_N^{p(\cdot)}(\Gamma,\varrho)$.
