General algebraic-form conjecture for generating functions counting occurrences of 321

Let $s_n((3,2,1),r)$ denote the number of permutations of length $n$ containing exactly $r$ occurrences of the pattern $(3,2,1)$, and let

$$F_{(3,2,1),r}(x)=\sum_{n\geq 0}s_n((3,2,1),r)x^n.$$

For each $r$, let $P_r$ and $Q_r$ denote polynomials. The 321 generating-function form conjecture. For every $r$,

$$F_{(3,2,1),r}(x)=\frac{1}{2x^{2r+1}}\left(P_r(x)+\sqrt{1-4x}\,Q_r(x)\right).$$

This conjecture proposes a uniform algebraic form extending the displayed cases. The paper contrasts it with a corresponding result for the $(3,1,2)$ pattern, which it says was proved by Bóna; the supplied text gives no proof or resolution for the 321 assertion.