Upgraded to 7.x-4.9. What is the correct way to upgrade library?

I've recently upgraded the MailChimp module to 7.x-4.9 (using drush up).

The release notes → says:

If upgrading to 7.x-4.9, please be sure to also upgrade to the latest version of the MailChimp library to ensure things go smoothly.

Following the link, the library release page says:

v1.0.8-package.zip contains the library and dependencies normally installed via Composer.

I have v1.0.6 of the library, and I want to "ensure things go smoothly". The question is how do I upgrade via Compsoer.

Both the project directory and the library directory has composer.json-files (those that ws installed when the library and module was installed).

Running composer upgrade in the Mailchimp project directory says "Nothing to install or update".

Running composer upgrade in the Mailchimp library directory says "Nothing to install or update".

There are no error messages.

How is this supposed to work?

I understand that I can download the library zip file and install it manually, but I thought the whole point of having a dependency manager such as composer was that it handled this for you.

So: What I am doing wrong? And how can I get composer to handle the MailChimp dependencies?