In my experience writing out assumptions together with estimates tends to improve estimate accuracy a lot, by challenging the assumptions (and estimates) more a

In my experience writing out assumptions together with estimates tends to improve estimate accuracy a lot, by challenging the assumptions (and estimates) more at estimation time, and making it more visible when changes occur at implementation time that impact estimates. As for negotiating estimates down, it is up to technical leadership to refuse to do that. Not easy, but doable.