According to the modern day ethnicity of Hungary, the largest population is German...
They must have been in the HRE if they were the leader of the HRE...
As said the HRE was an alliance of member nations, so if one is being attacked by a foreign power, then the others will assist.
Given that during the Mongol Invasion there were minor skirmishes between them and Venice and them and Bohemia, it is clear the the HRE did have involvement in the war on Mongolia...