Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open?

Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open?

Motivation: The Frattini subgroup of a profinite group by definition, is the intersection of all maximal proper open subgroups of the group. So I am asking of existence of maximal subgroups (between all subgroups) which are not open. Some of properties of such maximal subgroups are mentioned below in the comments:

1. They are not normal.

2. They are not closed. They are dense in the group.