From what I understand, the difficulty required by the proof-of-work in bitcoin is a function of block history, specifically the average time between last ~2000 blocks, with the intention that its hovers around the 10 minute mark.
The function relies on the timestamp of when the block was created. It occurs to me that when nodes are calculating blocks, they have an incentive to overstate the timestamp (because the difficulty would then be recalculated to allow for quicker mining of new blocks, which would lead to a higher rate at which that node receives rewarded bitcoins).
The protocol states nodes should reject blocks that have timestamps that look invalid (e.g. too far into the future), but again it relies on nodes being honest.
In short, what is the incentive for miners to not overstate timestamps? Am I right in thinking that if a majority of miners/nodes colluded in overstating + not rejecting invalid timestamps, then the entire bitcoin network would start generating bitcoins faster?
Note: It occurs to me that as the total pool of bitcoins increases + rate of reward halves every 4 years, this 'speeding up' of the bitcoin block generation has less and less impact.