Open, the site's label (page last edited 11 April 2026). No source resolves whether some $k$ exists. What is settled is that $k=3$ does not suffice: infinitely many even integers cannot be written as a prime plus at most three powers of $2$, by Crocker's 1971 construction [Cr71] and a parity argument, Lean-verified as the catalog variant Erdos10.erdos_10.variants.grechuk and accepted by the bounty site Conjectures.io on 6 August 2026 (see Current assessment); hence any such $k$ is at least $4$. This settles only the site's parenthetical remark, not the question. The variant has two claim pages, both partial: the site's certified record, an accepted partial claim, the certified Conjectures.io proof, and a second Lean proof of the same statement, listed on the forum, Daryxx 2026; no claim page settles the question itself, so the standing stays open. Search scope, dated 2026-09-28 UTC (the community database as of 2026-09-27): erdosproblems.com (the page, last edited 11 April 2026 and marked OPEN; the discuss and proof-claims forum threads, the latter holding one partial claim of 2026-08-07; the edit history), the community database (open, last update 2025-08-31), conjectures.io (the results listing, records 244ff2d0-399d-4e37-a307-4ff6f3cb3493 and ce95887b-8b61-4a89-9069-9131a58906e0, and the withdrawn problem page), the formal-conjectures commit history of FormalConjectures/ErdosProblems/10.lean, OEIS A387053 and arXiv:2605.17825; not searched: X, MathOverflow, Google Scholar and journal sites beyond the DOI check.