Regularity theorem for totally nonnegative flag varieties

We show that the totally nonnegative part of a partial flag variety $G\mathrm{/}P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.