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Claim. With $D(N)=\max_{0<a<N}D(a,N)$ the least possible largest denominator of a distinct unit-fraction representation of $a/N$, as Problem 305 defines it, the paper proves two bounds (as recorded on the library's source card). Theorem 1 (p. 602): for every $N$,

$$D(N)\le\lambda^3(N)\,N(\ln N)^2,\qquad \frac2{\log2}\ge\lambda(N)\ge1,\quad \lambda(N)\to1,$$

so that $D(N)\le(1+\varepsilon)N(\log N)^2$ for every $\varepsilon>0$ and large $N$, the form the zbMATH review (Zbl 0336.10007) gives. Theorem 4 (p. 612): for a prime $P$ large enough that $\log_{2r}P\ge1$, with $\log_j$ the $j$-fold iterated logarithm,

$$D(P)\ge\frac{P\log P\,\log_2P}{\log_{r+1}P\prod_{j=4}^{r+1}\log_jP}.$$

Covers. The prime lower bound, sharpened beyond the first paper's (its claim page): $D(b)$ is at least of order $b\log b$ on the primes, so the exponent $1$ of $\log b$ in the question cannot be lowered. Not covered: the upper bound the question asks for; the exponent $2$ of Theorem 1 does not reach $1+o(1)$, which Yokota's theorem (its claim page) does. The paper's introduction (p. 598) conjectures that the exponent $2$ can be replaced by $1+\delta$, the question itself.

Attribution. The site's commentary attaches this paper's exponent-2 bound to the first paper's key [BlEr76]; the problem page records the collision.

Acceptance. Refereed: M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), no. 4, 598--613. The publisher's record dates the issue 1 December 1976. 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.