Does disjunction have scope?

....trying to return to simpler questions today.

Case 1. Suppose Mary says,

Mary: I want pizza!

Later, Joe reports her utterance by saying

Joe: Mary wants pizza or cheesecake.

Is what Joe said true? My instinct is that it depends on some scope distinctions in the LF of what he said. On one reading (the wide scope one), what he said is correct: for it is true that [Mary wants pizza] or [Mary wants cheesecake]---it's true because the former is true. However, the narrow scope reading--Mary wants [pizza or pasta]--which is really

Mary wants [Pro has pizza or pasta].

seems false to me. Why? Simply put, because this is only true in Case 2:

Case 2. Mary says,

Mary: I want pizza or cheesecake.

Reporting this, Joe says,

Joe: Mary wants pizza or cheesecake.

Now this seems correct on the narrow-scope reading. (Is it correct on the wide-scope reading? Perhaps. I'm really not sure. I think I should probably say "yes.") The scopes should "match": if the disjunction occurred in what Mary said, then the disjunction should take narrow scope with regard to Joe's attitude report. However, if the disjunction is added by Joe as a result of his faulty memory ("...but I can't remember which") then the disjunction takes scope at the level of the attitude report--hence, wide.