I can see how Geseko and Gesko could be pet forms of some masculine name starting with Ger-, but is it really possible for them to be forms of
Gertrud? Did the culture of medieval Germany allow for the "masculinization" of basically feminine names?
Gesko and Geseko don't occur in J.
Van Der Schaar's _Woordenboek van Voornamen_. He does have Gesinus and under that Geeske, and seems to be saying (if I am understanding the Dutch properly) that it is a Latinized form of either names that began with Ger- or Gise-.