  Let $a$ be a semi-almost periodic matrix function with the almost periodic
representatives $a_l$ and $a_r$ at $-\infty$ and $+\infty$, respectively.
Suppose $p:\mathbb{R}\to(1,\infty)$ is a slowly oscillating exponent such that
the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue
space $L^{p(\cdot)}(\mathbb{R})$. We prove that if the operator $aP+Q$ with
$P=(I+S)/2$ and $Q=(I-S)/2$ is Fredholm on the variable Lebesgue space
$L_N^{p(\cdot)}(\mathbb{R})$, then the operators $a_lP+Q$ and $a_rP+Q$ are
invertible on standard Lebesgue spaces $L_N^{q_l}(\mathbb{R})$ and
$L_N^{q_r}(\mathbb{R})$ with some exponents $q_l$ and $q_r$ lying in the
segments between the lower and the upper limits of $p$ at $-\infty$ and
$+\infty$, respectively.
