  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
$\mathsf{Pop}_{M}: M \to M$ 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 $\mathsf{Pop}_{M}(M)$
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
\begin{equation*} \mathsf{Pop}(M; q) = \sum_{b \in \mathsf{Pop}_{M}(M)}
q^{|\mathscr{U}_{M}(b)|}, \end{equation*} where $\mathscr{U}_{M}(b)$ is the set
of elements of $M$ that cover $b$.
