Divergence everywhere of subsequences of partial sums of trigonometric Fourier series

It is proved that for any increasing sequence of natural numbers $\{m_j\}$ and any nondecreasing function $\varphi\colon[0,+\infty)\to[0,+\infty)$ satisfying the condition $\varphi(u)=o(u\ln\ln)$ ($u\to\infty$) there is a function $f\in L[0,2\pi]$ such that
$$ \int_0^{2\pi}\varphi(|f(x)|)\,dx<\infty, $$
and the Fourier partial sums $S_{m_j}(f)$ diverge unboundedly everywhere.