No terrain height is known (computed and baro ref) but the rad alt will come alive and display height at speeds at or above 250kts, egpws would not exist w/o it to further prove the point.
Here is wiki ref for egpws, it would not function in all modes without radalt. Certainly a heavy jet is not climbing out or decending over terrain at speeds less than 195kts.
The wikiref has refrences to solid publications that should end your dispute.
Sorry I am not allways a clear as I could be, I hope I was clear enough this time.
Ground proximity warning system - Wikipedia, the free encyclopedia