Fan, Guo, Su and our own Rui Xiong introduced "periodic pipe dreams" to compute fundamental classes of positroid varieties, as classes on Grassmannians. Their formula is too beautiful to live only in a quotient ring. I'll explain some subtleties in interpreting it geometrically upstairs in the polynomial ring, and extend it to compute classes of other projected Richardson varieties. As in the work of Fan, Guo, Su and Xiong, we will also discuss CSM classes of the related open varieties. This work is joint with Paul Zinn-Justin.