The image of the pop operator on various lattices

Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice M, a map ${\mathrm{Pop}}_M:M\to M$ that sends an element $x\in M$ to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study ${\mathrm{Pop}}_M\left(M\right)$ when M is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $B_n$. In particular, we settle four conjectures proposed by Defant and Williams on the generating function$\mathrm{Pop}(M;q)=\sum _{b\in {\mathrm{Pop}}_M\left(M\right)}{q}^{\left|U_M\right(b\left)\right|},$ where $U_M\left(b\right)$ is the set of elements of M that cover b.