The Proposition in the post is almost equivalent to the Conjecture, namely it implies that
$$\frac{g_n}{\log g_n}\leq (2+o(1))\log n.$$
In particular, the Proposition (hence also the Conjecture) implies that
$$g_n\ll\log n\,\log\log n.\tag{$\ast$}$$
On the other hand, it is expected that $g_n\gg (\log n)^2$ holds for infinitely many $n$'s, in which case $(\ast)$ is false. On the other hand, we don't know that $(\ast)$ is false. The best result in this direction is due to [Ford-Green-Konyagin-Maynard-Tao (2014)][1], and it states that
$$g_n\gg\frac{\log n \,\log \log n\,\log\log\log\log n}{\log \log \log n}$$ 
holds for infinitely many $n$'s.

  [1]: https://arxiv.org/abs/1412.5029