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Claim. The answer to Problem 516 is yes for every entire function $f(z)=\sum_{k\ge1}a_kz^{n_k}$ whose exponents satisfy $n_k>k(\log k)^{2+c}$ for some $c>0$. T. Kővári, A gap-theorem for entire functions of infinite order, Michigan Math. J. 12 (1965), no. 2, 133--140, doi:10.1307/mmj/1028999302, proves that for every entire function $f$ with strictly increasing exponents $n_k$ such that

$$n_k>k(\log k)^{2+c}\quad\text{for some }c>0\text{ and all }k,$$

one has $\limsup_{r\to\infty}\log m(r)/\log M(r)=1$, where $M(r)$ and $m(r)$ are the maximum and minimum modulus of $f$ on $\lvert z\rvert=r$; the theorem has no order hypothesis. The condition implies $n_k/k\to\infty$, so the finite-order functions it covers belong to the question's class. The statement follows the site's account of the paper and the formal-conjectures statement file for the problem, at the revision the problem page links, whose variant erdos_516.variants.limsup_ratio_eq_one states the theorem with strictly increasing exponents and cites the paper.

Covers. Entire functions of finite order with $n_k>k(\log k)^{2+c}$ for some $c>0$, a subclass of the question's class; the theorem itself covers every order. Not covered: finite-order functions with $n_k/k\to\infty$ and smaller exponents, settled by the accepted full claim of Fuchs.

Depends on. Nothing in this wiki: the argument is the paper's own.

Acceptance. Refereed: the paper appeared in the Michigan Mathematical Journal, a refereed journal. The site labels the problem PROVED (LEAN) and credits Fuchs with the solution, recording this paper's theorem as a result for arbitrary order; the label settles the problem through Fuchs's paper, not this one, so the page lists no reviewed evidence. The site records the conjecture that $\sum1/n_k<\infty$ should suffice in place of Kővári's condition; the problem page records a Lean counterexample posted to the thread against that conjecture as stated.

Dating. The page is dated by the issue date in the publisher's record (Crossref), 1 June 1965.