How do you fix the deadline and scope without estimates?

Within my teams, estimating doesn't appear to have downward pressure. Team members appear to want to estimate correctly because they want to have a realistic and steady workload. Driving down estimates means their workload goes up (at least in the short term, until our average velocity adjusts).

Estimates for anything sufficiently complex are pure fantasy. They just serve to place an engineer in a bargaining conundrum and makes them implicitly decrease the scope for the 1000's+ decisions they will need to make during the course of implementation that they are not going to deliver the desired (also implicit) expectations. Its better to work from deadlines, then at least the engineer can try to hedge down the scope and deliver something they are comfortable with.

And where are the deadlines coming from?

That's a people problem not a poker problem.