Konyagin's lacunary Walsh-Fourier convergence conjecture

Let $\{n_j:j\ge 1\}$ be a lacunary sequence of integers, meaning that

$$\inf_{j\ge 1}\frac{n_{j+1}}{n_j}>1.$$

For a function $f$ on $\mathbb T$, let $S_{n_j}f$ denote its Walsh-Fourier partial sums, and write $\log_+x=27+\max(0,\log x)$. Konyagin's conjecture. If

$$\int |f(x)|\log\log_+|f(x)|\,dx<\infty,$$

then

$$S_{n_j}f(x)\longrightarrow f(x)$$

for almost every $x\in\mathbb T$. This conjecture identifies the near-sharp integrability condition for almost-everywhere convergence along lacunary subsequences; the paper's main result comes close to resolving it, but the supplied text does not establish a complete resolution.