The shallow buried pipe will freeze and cause water backup to the downspout.
Where does the water drain to? Into the ground or does it exit above the ground at the end of the pipe.
If it exits, it may not freeze.
To prevent freezing, you could cover the pipe with 1 inch of blue styrofoam 12 inches to each side. This would prevent the frost from entering the ground.
Good Luck!
Dodgeman