How to get GNOME to recognize hibernate and make it available from dropdown menu and as a power button otpion?

Now that I managed to extend swap and enable hibernation, I'd like hibernate to be available from the menu and as an action under the power button.

I tried assigning hibernate through gsettings

$ gsettings set org.gnome.settings-daemon.plugins.power power-button-action hibernate

but it only brings up an interactive shutdown menu instead.

I have also tried looking for a GNOME extension to make hibernate visible in the menu. However, the only extension compatible with my GNOME (42.2) is Hibernate Status Button by p91paul which doesn't seem to have any effect.

How to enable hibernation with the power button and in the GNOME menu on Ubuntu 22.04?

I can successfully hibernate issuing

sudo systemctl hibernate

so the mechanism is working. I only need the UI options.