The cubic graph girth bound conjecture

Let $g(n)$ denote the largest girth among all cubic graphs on $n$ vertices. Girth bound conjecture. There is a constant $\epsilon_0>0$ such that

$$(2-\epsilon_0)\log_2 n \ge g(n).$$

The Moore bound gives only $g(n)\leq (2+o_n(1))\log_2 n$, while the best known lower bound is greater than $\frac{4}{3}\log_2 n-2$ for infinitely many explicitly constructed graphs. The conjecture proposes a uniform improvement to the leading constant in the upper bound.