Can Bitcoin(PoW) work under partial synchrony? This paper rigorously proves that Bitcoin can achieve consensus under the assumption of bounded message delivery time:


  Messages  in  a  mailbox may  be  delivered  in  any  order,  and  at 
  any  time,  except there is a maximum delay of ∆


If I understand it correctly, this only proves it works in synchronous network. If so, this assumption seems relaxed to me. 

Tendermint paper claims it can work under the partial synchronous assumption (from DLS paper):


  There is assumed to be some unknown upper bound ∆ on the time of
  messages to be delivered. 


The setup is more general in that the upper bound ∆ here is unknown. 

Can Bitcoin protocol achieve consensus in partial synchronous network as well? Is there any theory work dedicated to proving that?
If I understand it correctly, this only proves it works in synchronous network. If so, this assumption seems relaxed to me.


The Bitcoin network is not synchronous, it is asynchronous and maintains consensus.


  Can Bitcoin protocol achieve consensus in partial synchronous network as well?


Since Bitcoin is an asynchronous network and achieves consensus, it stands to reason that Bitcoin would be able to achieve consensus in a synchronous or partially synchronous network as well.