Rational generating-function conjecture for a non-Hertzsprung mesh pattern

Let $p$ be the mesh pattern represented by the diagram encoded in the source, and let $|\mathcal{S}_n(p)|$ denote the number of length-$n$ objects avoiding $p$. Rational generating-function conjecture. For this pattern $p$,

$$\sum_{n\geq 0}|\mathcal{S}_n(p)|x^n=\sum_{m\geq 0}m!\left(\frac{x}{1+x^2}\right)^m.$$

The question arises from a theorem giving this form for Hertzsprung patterns; the source presents this example as evidence that other nontrivial mesh patterns may also have such generating functions. The diagram-specific pattern notation is not defined in the supplied context.