Proved. The site records that Mrazović and Kovač, and independently Alon, observed that a suitable $Q$ exists: Vinogradov's three-primes theorem, in its distinct-primes form, gives an $N$ such that every prime above $N$ is a sum of three distinct smaller primes, so starting from the set of all primes at most $N$ every prime eventually appears in some $Q_i$. The two observations are the claim pages Mrazović and Kovač and Alon, accepted on the site's documented acceptance: the commentary of its curator, Thomas Bloom, who credits the observation to its authors, and the PROVED label. An archived copy of the page of 9 December 2024 already carries the credit and the label (then printed SOLVED), while one of 21 July 2024 shows the problem OPEN without it, so the claims are dated 9 December 2024. No paper exists and nothing was reviewed here. The thread's two comments of 1 January 2026 discuss explicit starting sets through Helfgott's proof of the ternary Goldbach conjecture and whether Ulam's set $\{3,5,7,11\}$ works; those concern a variant, since the question asks only for some $Q$. Erdős and Graham pose the problem, as Ulam's, on printed p. 94 of their 1980 monograph, together with a second Ulam problem on a greedy sequence of primes; the site's source key is [ErGr80, p. 94].