Lets make the charges on both plates the same $q,q$. Now ground one of them. What do you expect should happen? The charges on the grounded plate go into ground. Experiencing repulsion from the other like plate, they wish to get as far away as possible. Ground is like infinity, so they go there.
Conversely, when the charges on the plates are opposite, upon grounding, the charges on the plate would stay the same. In fact, instead of grounding, if you put another similar plate between the two and then connect it to any one plate, the charges on that plate would flow onto it.$^1$
In fact the above arguments hold for any pair of conductors in any charge config.
The charges on the plates of a capacitor aren't induced charges in the same sense as imparting temporary charge separation to a conductor via contactless electrostatic induction. That keeps the conductor neutrl. Instead the charging of each plate (while the whole thing stays neutral) is the $work$ of the charging battery. This is not to say that charges on one plate, are independent of the other when the battery has been removed. They bask in each other electric field.
But this point is irrelevant here. How charges got onto the two conductors has no bearing on what those charges will do when one of them is grounded.
$^1$ accompanied by energy loss.