Here it is: 
 ::::  
 ::: Sketch of proof that four colour changes are required: Suppose you try to get rid of the blue regions. Trouble is, each blue region touches all four other colours, the regions touching the inside blue one are disjoint from the regions touching the outside blue one. So in this case you need at least four changes. OK, so suppose instead you try to get rid of the red ones. (All other colours are symmetrical to the red ones, so handling red is good enough.) In this case there is a region touching both red regions (in fact you could pick either the yellow or the purple one), but it touches all three non-red colours. So here again, three changes aren't good enough. Have fun writing this more coherently ;-)