Run on host:

sudo visudo
pojntfx ALL=(ALL) NOPASSWD: /usr/bin/podman

Run inside of Toolbox:

sudo vi /usr/local/bin/podman
#!/bin/bash

executable=$(basename $0)
exec flatpak-spawn --host sudo "${executable}" "$@"

sudo chmod +x /usr/local/bin/podman

sudo vi /usr/local/bin/docker
#!/bin/bash

exec flatpak-spawn --host sudo podman "$@"

sudo chmod +x /usr/local/bin/docker

Now open a new session, and podman/docker should be running rootful even when executed from Toolbox!