Well, after half the afternoon, testing different things, her are my findings:
- USB-c monitor, or even the U-Green Hub (USB-A, USB-C data, no video, Ethernet Gigabit): randomly worked or not
- Uninstall “Extension Manager” v0.6.5 to avoid un-necessary headache
- Starting also afresh, without any Gnome-shell extension (except the system one: Ubuntu Dock and AppIndicator)
1/ After dozens of Hibernate / resume, to test in each possible configuration (battery only, power from original Framework charger, monitor on USB-C data only no power supply, USB-C u-Green hub plugged) I come to the conclusion that the Hibernation works randomly at the moment.
2/ I need to check 2 options in the /etc/default/grub file:
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash module_blacklist=hid_sensor_hub nvme.noacpi=1 resume=UUID=62e3f7fc-8d3c-45f7-b438-af31cf3b
What are:
I did not put them here when I set up Hibernation 2 years ago.
3/ I looked at the system logs, but there are too many things that I don’t understand to get a direction to look into.
I am posting them here if someone can have a look, please? You could spot something quickly 
Gnome-logs > Boot.log > export
From 21:34:53, Resume hibernation failed
From 21:35:52, Resume hibernation worked
21:35:52 systemd: Stopped target System Hibernation.
21:35:52 systemd: Stopped target System Hibernation.
21:35:52 systemd: Reached target System Hibernation.
21:35:52 systemd: systemd-hibernate.service: Consumed 1.173s CPU time.
21:35:52 systemd: Finished Hibernate.
21:35:52 systemd: systemd-hibernate.service: Deactivated successfully.
21:35:52 kernel: PM: hibernation: hibernation exit
21:35:52 systemd-sleep: System returned from sleep state.
21:35:52 kernel: PM: hibernation: Basic memory bitmaps freed
21:35:27 systemd-sleep: Entering sleep state 'hibernate'...
21:35:27 kernel: PM: hibernation: hibernation entry
21:35:27 systemd: Starting Hibernate...
21:35:09 gsd-media-keys: Failed to grab accelerator for keybinding settings:hibernate
21:34:53 kernel: PM: hibernation: resume failed (-1)
21:34:53 kernel: Hibernate inconsistent memory map detected!
21:34:53 kernel: PM: hibernation: Basic memory bitmaps created
Export Important messages:
21:35:52 bluetoothd: src/device.c:set_wake_allowed_complete() Set device flags return status: Invalid Parameters
21:35:09 gdm-session-wor: GLib-GObject: g_object_unref: assertion 'G_IS_OBJECT (object)' failed
21:35:07 systemd: Failed to start Application launched by gnome-session-binary.
21:35:07 gdm-session-wor: gkr-pam: unable to locate daemon control file
21:34:55 gnome-session-b: GLib-GIO-CRITICAL: g_bus_get_sync: assertion 'error == NULL || *error == NULL' failed
21:34:54 thermald: Unsupported conditions are present
21:34:53 bluetoothd: src/device.c:set_wake_allowed_complete() Set device flags return status: Invalid Parameters
21:34:53 avahi-daemon: chroot.c: open() failed: No such file or directory
21:34:53 kernel: cros-usbpd-charger cros-usbpd-charger.2.auto: Unexpected number of charge port count
21:34:53 kernel: Module hid_sensor_hub is blacklisted
21:34:53 kernel: PM: hibernation: Failed to load image, recovering.
21:34:53 kernel: Hibernate inconsistent memory map detected!
21:34:53 kernel: Module hid_sensor_hub is blacklisted
4/ I would like to add the syslog - partly filtered with 3 hibernate / resume cycles, but I don’t know how to attached it here (file type not supported), and it is very long. Any advice to post only what is useful would be appreciated 