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Suppose $\Gamma$ is a Carleson Jordan curve with logarithmic whirl points,  $\varrho$ is a Khvedelidze weight, $p:\Gamma\to(1,\infty)$ is a continuousfunction 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 generalizedLebesgue space with variable exponent. We prove that all semi-Fredholmoperators in the algebra of singular integral operators with $N\times N$ matrixpiecewise continuous coefficients are Fredholm on$L_N^{p(\cdot)}(\Gamma,\varrho)$.

Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm