Boros, Caro, Füredi and Yuster's asymptotic conjecture for non-repeated cycle lengths

Let $f_2(n)$ denote the maximum, over all $n$-vertex 2-connected graphs, of the number of cycle lengths that occur exactly once. The authors' conjecture is

$$\lim_{n\to \infty} f_2(n)/\sqrt{n} =1.$$

This conjecture asserts that the lower bound obtained from Sidon sequences is asymptotically tight. The known construction gives $f_2(n)\geq \sqrt{n}-o(\sqrt{n})$, while the matching upper bound remains open.