Proved. The answer is yes. Price posted an argument generated with GPT-5.2 Pro in the site's thread on 5 February 2026: a sieve count of the representations $2^k+m$ with $m$ free of primes in $(z,x^{1/t}]$ for large constants $z,t$, whose first and second moments the fundamental lemma of sieve theory estimates, the second moment resting on an averaged bound for a singular series over the primes dividing $2^k-2^l$. Tao confirmed the proof correct in an edit to his thread comment of 5 February 2026, and the site's curator, Thomas Bloom, labels the problem PROVED and credits the solution to Price (page last edited 2 April 2026, accessed 2026-09-05 and 2026-10-07; five comments, no proof claim, no exposition). No refereed or arXiv version was found on 2026-10-07. Romanoff (1934) gives the $r=1$ case with a positive density in place of $1-\epsilon$. Claim page: Price 2026 (accepted on Tao's confirmation and the site's curator's acceptance; not refereed, not counted as formalized). A thread comment of 6 February 2026 by Sawhney announces that he and Green have a version of the argument that gives $r\ll\log(1/\epsilon)/\log\log(1/\epsilon)$ by a high-moment argument in place of the second moment, which the site's commentary also mentions; it is a thread comment without a write-up, no manuscript was found on 2026-10-07, and it has no claim page.