Explicit generating-function form conjecture for even and odd 132-occurrence counts

Let $E_r(x)$ and $O_r(x)$ denote the generating functions for even and odd permutations, respectively, with exactly $r$ occurrences of $132$. For every $r\geq1$, there exist polynomials $A_r(x)$, $B_r(x)$, $C_r(x)$, and $D_r(x)$ with integer coefficients such that

$$\begin{aligned} E_r(x)&=\frac{1}{4}(A_r(x)+B_r(x))+\frac{1}{4}C_r(x)(1-4x)^{-r+1/2}+\frac{1}{4}D_r(x)(1-4x^2)^{-r+1/2},\\ O_r(x)&=\frac{1}{4}(A_r(x)-B_r(x))+\frac{1}{4}C_r(x)(1-4x)^{-r+1/2}-\frac{1}{4}D_r(x)(1-4x^2)^{-r+1/2}. \end{aligned}$$

Explicit generating-function form conjecture. The displayed representations hold for every $r\geq1$. The paper presents this as a stronger conjecture motivated by explicit results; no proof or disproof is supplied in the given text.