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Claim. With $D(a,N)$ the least possible largest denominator in a representation of $a/N$ as a sum of distinct unit fractions and $D(N)=\max_{0<a<N}D(a,N)$, as Problem 305 defines them, the paper proves two bounds. Theorem 1 (p. 158): for every prime $P$,

$$D(P)\ge P\lceil\log_2P\rceil,$$

with $\log_2$ the base-2 logarithm, which is the site's $D(p)\gg p\log p$. Theorem 2 (p. 162): there is a constant $K$ with $D(N)\le KN(\ln N)^3$ for every $N\ge2$. The zbMATH review (Zbl 0328.10010) gives the same two statements. The paper closes with the question itself as its Conjecture 3 (p. 167).

Covers. The prime lower bound: $D(b)$ is at least of order $b\log b$ on the primes, so the exponent $1$ of $\log b$ in the question cannot be lowered and the estimate $D(b)=b(\log b)^{1+o(1)}$ that the solution gives is sharp in the exponent along the primes. Not covered: the upper bound the question asks for. Theorem 2's exponent $3$, and the exponent $2$ of the sequel (its claim page), do not reach $1+o(1)$; Yokota's theorem (its claim page) does.

Attribution. The site's commentary credits the bound $D(b)\ll b(\log b)^2$ to this paper under its key [BlEr76]; the exponent-2 bound is Theorem 1 of the sequel in the Illinois Journal of Mathematics, and this paper prints exponent $3$. The problem page records the collision.

Acceptance. Refereed: M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions, J. Number Theory 8 (1976), no. 2, 157--168. The publisher's record dates the issue May 1976 and gives no day; this page is dated the first of that month. The site's PROVED label credits Yokota's paper, not this one, so no reviewed evidence is listed. This claim is partial: it settles the lower half of the estimate, not the question.