Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$

For a gap sequence of natural numbers $\{n_k\}^\infty_{k=1}$, for a nondecreasing function $\varphi\colon[0,+\infty)\to[0,+\infty)$ such that $\varphi(u)=o(u\ln\ln u)$ as $u\to\infty$, and a modulus of continuity satisfying the condition $(\ln k)^{-1}=O(\omega(n_k^{-1}))$, we present an example of a function $F\in\varphi(L)\cap H_1^\omega$ with an almost everywhere divergent subsequence $\{S_{n_k}(F,x)\}$ of the sequence of partial sums of the trigonometric Fourier series of the function $F$.