First, be sure that you're not logged into any other LinkedIn account in the browser where you're connecting the integration besides that of the account that you wish to connect. Note that you can still be logged in even if you don't have the LinkedIn site currently open.

Next, navigate to the campaign where you'd like to connect the LinkedIn integration, then click "Integrations" from the menu on the left side of the screen.

Click on LinkedIn in the list of available integrations.
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Then click Connect New Account.

A window will then pop up prompting you to log in to LinkedIn. Log in with an account that has Super Admin access to the LinkedIn page that you are trying to connect to.

You will then be prompted to select the customer account which you would like to connect. Click the radio button next to the appropriate account, then click "Save".

If the account that you would like to connect is under a different LinkedIn account, and you need to enter different credentials to access that account, click the button that says "Connect New Account", then enter the appropriate credentials, and step through these screens again.