For Solus Budgie, you need to create the /etc/lightdm/slick-greeter.conf file whose content is the full path to the image to use. (Replace /home/robg/Pictures/mywallpaper.jpg with the full path of the image you want to use.)
If /etc/lightdm doesn't exist, you need to create it with sudo mkdir /etc/lightdm.
This is what Joshua Strobl suggested in Changing the Login screen wall paper; it works also for Solus Mate.
Alternatively, you can do as suggested in How to change the login screen wallpaper? Install Login Window with sudo eopkg install lightdm-settings, and execute it by the application menu. It will show a window where you can choose the background image for the login screen.
Remember to deselect Draw user backgrouds. Also, since you will be asked to authenticate, the directory where the images are initially looked is the home directory for the root user. To select an image in your home directory, you need to select Other Locations in the file dialog, and then Computer; from there you can select the home directory, which contains your home directory.
I would rather follow the first suggestion, if the background image is the only thing you need to change, and you don't change it often.