My information was the Himmler kept the rumor of Heydrich’s Jewish blood as a way to keep him ardent in his work.
A “make him prove his is not” by being “extra zealous.” Hitler used that tactic too, on some of his “justice ministers.”
So I have been told.
Reinhard Heydrich’s father, the composer Richard Bruno Heydrich, had been effectively accused of being Jewish when a dictionary of German music listed him as having changed his name from a Jewish one. Apparently this wasn’t true but it became an obsession of the father to “clear his name” and perhaps this was passed on to the son.