They’re already in conflict everywhere. Infrastructure for cars robs public transit infrastructure blind in lots of government budgets. The only public transit category potentially benefiting from car infrastructure is buses, which are arguably the worst form of public transit to begin with, and still also require additional dedicated infrastructure to get any better (e.g. dedicated bus lanes).
“Self-driving” cars obviously require car infrastructure which already steals from public transit budgets both federally and locally, but if we add government emphasis on this technology and start to develop specific infrastructure for “self-driving” cars (walled off routes, communications appliances, etc.) then they’ll start taking even more of the budget.
And all of this for something that’s arguably much more braindead and useless and consuming of R&D dollars than the obviously more efficient, already technically possible forms of transit that could be built or expanded upon today.