Environment variables only set with specific display managers

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Using Hyprland (on Wayland with greetd + ReGreet as display manager) and wezterm or KeePassXC as example applications, the SSH_AUTH_SOCK variable is not accessible to the applications. I have it in inteactive zsh though, as echo $SSH_AUTH_SOCK returns the usual /run/user/1000/ssh-agent.
With SSDM as a display manager, the SSH_AUTH_SOCK variable is accessible in zsh and KeePassXC, for example.
I start ssh-agent not manually, but with home-managers services.ssh-agent.enable = true;. In both cases, it is running and fine, as ssh-add works. My configurations (only excerpts, if more needed will provide):

# home/hyperland.nix
    wayland.windowManager.hyprland.enable = true;
    wayland.windowManager.hyprland.settings = {
      # ... other configuration ...
      env = ["XCURSOR_SIZE,24" "HYPRCURSOR_SIZE,24"];
      exec-once = [
        # "systemctl --user import-environment SSH_AUTH_SOCK"
        "[workspace 1 silent] wezterm"
        "[workspace 10 silent] ${lib.getExe' pkgs.keepassxc "keepassxc"}"
      ];
};

# system/graphical.nix
    services.greetd.enable = true;
    services.greetd.settings = {
      default_session = {
        command = "${pkgs.dbus}/bin/dbus-run-session ${lib.getExe config.programs.hyprland.package} --config /etc/greetd/hyprland.conf";
        user = "greeter";
      };
      terminal = {vt = 1;};
    };
    environment.etc."greetd/hyprland.conf".text = ''
      # exec-once = systemctl --user import-environment SSH_AUTH_SOCK
      exec-once = ${lib.getExe config.programs.regreet.package}; ${lib.getExe' config.programs.hyprland.package "hyprctl"} dispatch exit
    '';
    programs.regreet.enable = true;
    programs.regreet.settings = {
      background = {
        path = ../../.config/pictures/wallhaven-pastel-mountain.png;
      };
    };
    programs.hyprland.enable = true;