Yes you do 
The item has to store something in a variable in the GI so that the player knows the pickup has happened. Otherwise how do you know in the next level?
What you put in the GI is up to you and depends on how many different sorts of things the player can pick up.
Let’s make it very simple to start with, just a bool. So the pickup sets the bool in the GI before it destroys itself. In the new level, the player reads the GI, sees the bool is true, and puts the pickup in their hand again.
Like that.