Erdős problem 996

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$ with $n$th Fourier partial sum $f_n$. Is there an absolute constant $C>0$ such that if $\| f-f_n\|_2 \ll (\log\log\log n)^{-C}$ then $\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f$ for almost every $\alpha$? A preprint answers this negatively via a dyadic spike-block counterexample.