Erdős problem 285

Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \dots < n_k$ with $$1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}.$$ Is it true that $$f(k) = (1 + o(1)) \frac{e}{e - 1} k ?$$