Try archiving conversations on LinkedIn mobile app:
If you do not want to delete any conversations then you can archive that instead of deleting it on your LinkedIn's mobile app. This way the conversations are moved to the Archived folder.
Archiving on a LinkedIn iOS app:
Open the Messaging tab on your LinkedIn app.
Then find the conversation that you want archived and swipe left on it.
Now, tap Archive.
Now, you have changed your mind and want to unarchive the conversations then:
In the LinkedIn app, find the Messaging tab and on the top right tap on Filter.
Find Archived in the menu and tap on it.
And for you to view the message tap on it.
And unarchive the message by swiping left on it. Then tap Restore.