Let $e_6(n)$ be the greatest number of edges in a simple graph with $n$ vertices and girth at least 6. Let $G_6(n)$ be the set of simple graphs of order $n$ with girth at least 6 and $e_6(n)$ edges.

My question: Is there any $n$ for which none of the graphs in $G_6(n)$ is bipartite?

From computer experiments, I have found that the only values of $n\le 47$ for which $G_6(n)$ has any non-bipartite graphs at all are 7 (7 edges), 9 (10 edges), 15 (22 edges), 27 (53 edges), and 43 (106 edges). However, in all those cases $G_6(n)$ includes bipartite graphs as well.