For possible canon sources we have:
- The original TV series
- The original comic series
- The Return of the Joker (DVD)
In the original TV series Batman (Terry) travels to the Fortress of Solitude in the The Call 1 & 2. And yet in the Batman Beyond Unlimited #2 he says this while conversing with Superman.
I would think he already knows that Superman has a sort of trophy room. It is within the Fortress of Solitude.
In the same comic issue Batman (Terry) refers to how Tim Drake ended his crime-fighting career originally. Is he referring to the events of The Return of the Joker? Or something else?