Much of the hypothetical reasoning here comes from Wiard Hinrichs.
In a trial before the Reichs court of appeal in Speyer, 30 July and 13 October 1587, appearing as a witness we find (according to Gretje Schreiber, in Heim und Herd 6, 17 July 1993, page 24) Tiarck Gaten of Osteel, 52 years old (thus born 1535), “ernehre sich seiner Ochßen und Koyen [defending? his oxen and cows].” He could have had a son named Gate Tjarks.
But this son must have been born by the late 1550s in order to have been a father in 1581.
For further details on this line of conjecture, and on the rest of the family, see Kay Blass’s notes on Tjerck Claessen De Witt.