Open, the site's label as accessed. No result settles the question. The OpenAI release's superexponential lower bound $W(k)>k^{k/100000}$ for all large $k$ (23 September 2026), recorded on the claim page OpenAI 2026 as a claimed partial result, would answer the case $c=1$ in the negative through the identity above; the bound is accepted, formalized, on Problem 138, and the reduction is elementary but has not been checked by review or formalization here. The case $0<c<1$ is open. Lean developments posted in the site's thread in June 2026 claim yes answers, with polynomial bounds, to the questions $N(k,2)\le C^k$ and $N(k,\sqrt k)\le C^k$; they are recorded as claimed partial results on Kitamura, 21 June 2026 and Kitamura, 23 June 2026. For even $k$ the first already follows from Spencer's formula for $N(k,1)$, since $N(k,2)=N(k,1)$ by parity. No full claim exists, and the standing derives from the claim pages.