I have been able to get this working on my Framework 13 11th gen (currently 12th gen).
Here are the details for getting it up and running. Basically when you install fprintd and fprint-pam (not sure about the spelling on this, but guides will make it clear. I am just typing this up fast) the service will not autostart as it is supposed to.
This is because a dbus connection is not created. To fix that do the following:
sudo nano /usr/share/dbus-1/system-services/net.reactivated.Fprint.service
then put the following in the file:
[D-BUS Service]
Name=net.reactivated.Fprint
Exec=/usr/libexec/fprintd
User=root
SystemdService=fprintd.service
Save and exit.
Now we need to restart dbus:
sudo systemctl reload dbus
This will get it working as it should. To verify you can open system settings and then go to users. You should see an option there to configure the fingerprint scanner.
If you would like to get the fingerprint scanner working in the terminal do the following:
Make a copy of pam.d sudo file:
sudo cp /etc/pam.d/sudo /etc/pam.d/sudo.bak
Then open it and edit it:
sudo nano /etc/pam.d/sudo
add the following line to the top of the file:
auth sufficient pam_fprintd.so
such that your file looks similar to this:
auth sufficient pam_fprintd.so
auth include system-auth
account include system-auth
password include system-auth
session include system-auth
Save and exit (Ctril + O and Ctrl + X)
As you have just used sudo, you should close the terminal and reopen it. Then try the following:
sudo -i
If everything is good, it will prompt you for your fingerprint. (This assumes you set one up when you were checking it in system settings under users.)
For me this means that Open Mandriva works perfectly on the Framework 13.