plural of leur médaille

Bonjour John B.

I don't question your logic, but my thought is that each competitor receives only one medal. 

Yes, the group as a whole receive medals but not individually.

Hope this is helpful.

Bonne journée

Jim

John, 

for further information and discussion, see especially ’ the really tricky part ‘ in the attached link from Laura Lawless.

Describing  multiple possessors with multiple possessions in French - if only one possession for each possessor or the ‘group’, the singular ‘ leur ‘ is needed; if each possessor has multiple medals, the plural ‘ leurs ‘ is needed.  

Here the issue is that each person could have one or more medals, but it is unlikely that each person will indeed have more than one medal. 

Even though English approaches from the possessor side of things, it also frequently leaves uncertainty when there is a combination of plural possessions and plural possessors.

 https://www.lawlessfrench.com/grammar/possessive-adjectives-plural-possessions/