The product formula conjecture for pinnacle-set enumeration

For a finite set $S$ of pinnacle values, let

$$q_n(S):=\sum_{I\subset S}2^{|I|}p_n(I).$$

Write $S=\{s_1,\dots,s_p\}$ with $n\geq s_p$, define

$$q'_n(S)=2^{n-1}2^{s_p-2}2^{-s_{p-1}}2^{s_{p-2}-2}2^{-s_{p-3}}\dots,$$

and set

$$D(S)=(d_1,\dots,d_{p-1}):=(s_1-1,s_2-s_1,\dots,s_{p-1}-s_{p-2}).$$

Define abstract expressions by $E_1=x_{1,3}+x_{1,1}$, $E_{2k+1}=f_e(E_{2k})$, and $E_{2k}=f_o(E_{2k-1})$, where

$$f_o(x_{a,k})=x_{a,k}\left(x_{a+1,\frac{k+1}{2}}+x_{a+1,\frac{k-1}{2}}\right),$$

with $x_{a,0}=0$, and

$$f_e(x_{a,k})=x_{a,k}(x_{a+1,2k+1}+x_{a+1,2k-1}).$$

If $S$ has $p$ elements, define $r_n(S):=\operatorname{ev}(E_{p-1})$, where $\operatorname{ev}$ evaluates $x_{i,j}$ to $j^{d_{p-i}}$. The conjectural product formula. For all pinnacle sets $S$ and all $n\geq\max(S)$,

$$q_n(S)=q'_n(S)r_n(S).$$

This conjecture gives a general formula for the weighted enumeration $q_n(S)$ of permutations associated with pinnacle sets; the paper presents an algorithmic factorization and initial explicit cases, but no resolution is supplied here.