The theoretical complexity of Go - The state of the art
The state of the art for the theoretical complexity of go is well summed up on Wikipedia, with relevant references.

The main remaining open problem is for rules using a superko, i.e. repeating any past position is forbidden. This is the rule used for instance in China  and western countries. It is simple to state, but could bring some difficulties to actually play in some situations, as players need (in theory) to remember all the history of the game. In reality, these situations are so rare that it does not matter. For this rule, as you mention, we only know that deciding the winner from a given position is PSPACE-hard and in EXPSPACE.

On the other hand, it is known from Robson that Go with Japanese rules is EXPTIME-complete. These are the rules using local ko, only forbidding to repeat the last  position, and declaring draw if a longer cycle occur. These rules have to include many special cases to settle the situation for particular configurations, and although they are considered more practical for human players, especially for beginners to learn about the ko rule, they are not elegant mathematically.

You can see a sketch of Robson's proof that Go with Japanese rules is EXPTIME-hard here, with fun go positions using ladders to encode boolean formulas.