I guess the closest you can come to this is by taking a screenshot of your desktop and setting it as the background of the display manager. Otherwise, Hyprlock seems like your best option.

DankMaterialShell does this with its lockscreen however i dont know if the greeter behavies similarly nor if it has auto login

what you are asking for doesn't really make sense.

a display manager is for running when no user is logged in. its purpose is to log in and start your session. thats why they can't show a desktop as a background because ot isn't running yet. once you log in the display manager quits because its job is done.

u could use an custom bash file that screenshots ur desktops, blurs it, and then sources it as ur lockscreen from a config file, in bash it could go:

#!/usr/bin/env bash

bg=~/.lockscreen.png
grim -o $bg
convert $bg -blur 0x8 $bg 
hyprlock
rm $bg

havent tried it yet, but as long as ur sourcing the ~/.lockscreen.png on ur config file, or u launch ur locking program with an argument to use whatever u used as a background (and that way u can use mktmp instead of throwing it on ur home folder) it should work. Just try the commands b4 executing it since i wrote this from my bed on a saturday and my brain is half working.

i just used hyprlock as an example, since u said ur using hyprland, but u can use whatever ofc, I myself use swaylock.

You may screenshot the desktop after login and use it as a background, but I don't think it will be exactly the same

Youre looking for a lockscreen. like people have suggested, id say the sweetest way is writing your own simple script. For example, take a look at my lockscreen script to get a feeling of things.

your can take a screenshot of the screen and blur it and set it as background each time.

https://codeberg.org/alirezaalavi/dotfiles/src/branch/main/sway-config/.config/scripts/swaylock-custom

Display managers handles sessions. A "desktop" is a session and if none are started yet, how can it display it?

Lockscreen and display managers are two different things. They are not the same and does not have the same functionality.

When you log out, have a script take screenshot, blur it and save to a known file.

Conf your display manager to use that file as background.

I dynamically take a screenshot and blur it

That's a compositor's job. Look at what compositor you have and how to enable this.

I vaguely remember doing something like this a long time ago. I changed my screen lock script to take a screenshot of my desktop, then with ImageMagick scale the image to something like 10% then back up to blur it, then save it to overwrite the lock screen background.

Dropping to the display manager would kill your desktop session, as far as I know. I think what you maybe want is Hyprlock. Setup whatever style you want, and than spawn both the Hyprlock daemon and activate it on login.