Vilenkin-Fourier series in variable Lebesgue spaces

Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f\in L^1(G)$. For $1<p_{-}\le p_{+}<\mathrm{\infty }$, we characterize all exponents $p(\cdot )$ for which the convergence of $S_{n}f$ to $f$ in $L^{p(\cdot )}(G)$ holds whenever $f\in L^{p(\cdot )}(G)$.