Proved. The asymptotic $k(N)\sim2N^{1/2}$ was first proved in 1995: Theorem 1 of Deshouillers and Freiman (Israel J. Math. 92 (1995), 33--43) gives $k(N)\le2N^{1/2}+CN^{5/12}$, which with Straus's block is $k(N)=2N^{1/2}+O(N^{5/12})$, the affirmative answer to the displayed question. The exact value for large $N$ followed in 1999: for all $N\ge N_0$ ($N_0$ effectively computable, not made explicit), Theorem 1 of Deshouillers and Freiman (Astérisque 258 (1999), 141--148, published by the Société mathématique de France, whose Crossref record types the article as a journal article in Astérisque) gives $k(N)\le2\sqrt{N+1/4}-1$, and Straus's block $\{N-k+1,\ldots,N\}$, admissible exactly when $k\le2\sqrt{N+1/4}-1$ (as the same paper reports and as Erdős, Nicolas and Sárközy state in the form $k=2m-1$ for $m^2\le N<m^2+m$, $k=2m$ for $m^2+m\le N<(m+1)^2$), attains it; so $k(N)=\lfloor2\sqrt{N+1/4}-1\rfloor$ for $N\ge N_0$, hence $k(N)=2N^{1/2}+O(1)$ and $k(N)\sim2N^{1/2}$, the affirmative answer. The combination of the theorem with the block is a one-line deduction made here. The earlier bounds $\limsup k(N)N^{-1/2}\le4/\sqrt3$ (Straus, reproved in 1991) and $\le(143/27)^{1/2}$ (Erdős, Nicolas and Sárközy, Théorème 1) and Erdős's $CN^{5/6}$ (1962) are superseded and answer neither part of the question. Straus's paper and Erdős's 1998 paper, both site keys or sources, are not held; for small $N$ the equality $k(N)=$ block size is the numerical conjecture of Erdős, Nicolas and Sárközy, not a theorem. The two accepted claims are recorded on the claim pages of the 1995 paper (refereed; the site credits the 1999 paper, not this one) and the 1999 paper (refereed, with the curator's credit).