MJUK

James thanks for your comment about QQ+/AK+ as being the best scenario there.  Given you opened in LP. Couldn't his perception be you are quite wide?  If you originally opened UTG or UTG+1 i would agree but you opened LP get 3bet then the BB is priced in fairly wide.  At this stage he knows you still wide and your 4 bet could still have lots of AQo AJs hands in your range given the action.  There is nearly 100 in the pot when he shoves and i think lots of players will do this atleast as wide as QQ+ but given the action JJ as well.

4-bet folding KK just seems wrong.  Even if you assume it is QQ+/AK then you have 52% equity so it is a call with the money already in the pot.