This sounds like an outright BS claim. There are two or more scenarios that may be presented as evidence.

i. LinkedIn used the users current passwords with their external addresses to access the external emails. ( impossible) ii. Linked in use some sort of Oauth/google authentication access to information permission thing(can't remember the name). (highly unlikely)

In any case I think we can only be certain with the actual evidence.

The Customers filing suit should know that LinkedIn is a publicly traded company and not a scam site.

Because even these claims are outrages if not utter BS.