It makes no difference who escalates the dispute....paypal allows the other party 10 days to respond before going to the next step.

If the other party does not respond then the dispute will close in your favour.  If they respond then it is game on and you are given the opportunity to respond again.

If they respond in less than the 10 days you can try ringing paypal and ask for a resolution if time is of the essence but there is no guarantee that they will oblige.  I have had disputes drag on for 3+ weeks.