So you see a full complement of your contacts in the Contacts app at murena.io?
On the phone,
Settings > Accounts … is expected to show your Contacts account (usually as the second entry). This is expected to cause the murena Contacts to sync with the Contacts app on the phone.
If your FP6 Contacts app is different from murena Contacts app, there must be an account sync problem.
Sync can be adjusted and Sync now buttons can be found within the Accounts page. Is the problem covered there?
The dialer will rely on FP6 Contacts app in the first instance.