Lifted veneer on old dresser

I received an old dresser from a neighbor who was moving. It had belonged to his mother and had been sitting on a family member's back porch where it was exposed to the elements. It's in reasonably good condition except for the top where the veneer has lifted right at the front edge. I've never dealt with anything like this and would be grateful for any suggestions as to how to repair it. It's a lovely little dresser and I'd like to keep it.
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