DISPROVED (LEAN). The status-defining source is Stijn Cambie's observation on the site's thread (2025-08-10): with $n_k=2^k$ and $a_k=2^{k-1}+1$ the only uncovered $m<n_k$ is $1$, so the count is the constant $1$ and the statement is false; the claim page is Cambie (accepted on the curator's credit; no refereed source). The site's Lean qualification refers to the Lean formalization recorded under Formalization.