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Maximal subgroups which are not open in pro-2 groups

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.