Pop polynomial formulas for weak, Tamari, Cambrian and root-poset lattices

Let $W$ be a finite Coxeter group. Write $\mathrm{Weak}(W)$ for its right weak order, $\mathrm{Tamari}(W)$ for a Cambrian lattice associated to a Coxeter element obtained from a linear orientation of a path Dynkin diagram, $\mathrm{Camb}_\mathrm{bi}(W)$ for a Cambrian lattice associated to a bipartite Coxeter element, and $J(\Phi^+_W)$ for the distributive lattice of order ideals of the positive root poset of type $W$. Let $\mathsf{Pop}(L;q)$ denote the pop polynomial of a lattice $L$. The pop-polynomial conjectures. The following equalities hold:

$$[q^{n-1}]\mathsf{Pop}(\mathrm{Weak}(B_n);q)=3^n-2n-1.$$ $$\mathsf{Pop}(\mathrm{Tamari}(A_n);q)=\sum_{k=0}^n\frac{1}{k+1}\binom{2k}{k}\binom{n}{2k}q^{n-k}.$$ $$\mathsf{Pop}(\mathrm{Tamari}(B_n);q)=\sum_{k=0}^{\lfloor (n+1)/2\rfloor}\binom{n-1}{k}\binom{n+1-k}{k}q^{n-k}.$$ $$\mathsf{Pop}(\mathrm{Camb}_\mathrm{bi}(A_n);q)=\sum_{k=1}^{\lfloor (n+3)/2\rfloor}\frac{k(-1)^{k-1}}{n-k+3}\sum_{j=0}^{n-k+3}\binom{j}{n-j+3}\binom{n-k+3}{j}q^{j-2}.$$ $$\mathsf{Pop}(J(\Phi^+_{A_n});q)=\sum_{k=0}^n\frac{1}{k+1}\sum_{j=0}^{n-k+1}\binom{k+1}{j-1}\binom{k+1}{j}\binom{n-j+1}{n-k-j+1}q^{k+1}.$$ $$\mathsf{Pop}(J(\Phi^+_{B_n});q)=(-q)^n+\sum_{k=0}^n\sum_{j=1}^{k}\binom{k+1}{j}\binom{n-k-1}{j-1}\binom{n-j}{n-k}.$$

These formulas conjecturally extend enumerative results for pop-stack sorting on weak orders, Cambrian lattices and order-ideal lattices; the displayed sequences agree with the cited OEIS data, but the source supplies no resolution of the conjectures.