Not totally true. My van does a regen when the fuel level is well into the red. Others on here also have commented that their vans also do.If the low fuel light is on it won't burn. But the carbon keeps growing, so not filling up enough to turn off the light allows carbon to go above the next threshold, somewhere around 40g, (But I've not recorded it). might be as high as 50g. Then when you do a big fill up the regen light comes on and it set the burn process in motion.
It won't burn with low water temparature.
Also mine can start a regen within a minute or two of starting the engine, particularly if the regen didnt finish because the van was left to idle in neutral until the van interrupted the regen.