I'm curious about the possibility that mobile phone radiation does not cause cancer, but its presence can increase the effectiveness of other things the can cause cancer.
There are some researchers who think that the mechanism cells use to detect and repair damage to DNA finds the damage by looking for places along the DNA where there are unexpected changes in electrical conductivity such as due to broken bonds. There has been speculation that passing non-ionization electromagnetic radiation could interfere with that mechanism, causing it to overlook damaged DNA that it would have otherwise found and repaired.
Note that if it does indeed have some kind of "suppress preventing cancer" effect, then the intensity of the radiation might not matter much as long as it is above some threshold. What might matter most is the length of continuous exposure, and the time between intervals of exposure. As long as the times of little or no exposure are frequent enough and long enough that the cells will be able to do their find and repair operations before a cell on its way to cancer has time to start reproducing, you should be fine.
This would differ quite a bit from the usual radiation testing situation, where all that matters is total exposure, so you can simulate long term low level exposure by short term, higher level exposure.
It could turn out, say, that 100 short calls spread throughout the day are almost completely safe, even on an old high power phone, but a couple of long conversations greatly increase the risks that if something else tries to give you cancer during that time it will succeed.
This would probably be pretty hard to conduct a study on.