The generating-function conjecture for valid hook configurations avoiding length-3 and length-4 patterns

Let $\operatorname{Av}_n(\sigma_1,\ldots,\sigma_r)$ denote the permutations of length $n$ avoiding the listed patterns, and let $\operatorname{\mathsf{VHC}}(\mathcal{S})$ denote the set of valid hook configurations associated with a permutation class $\mathcal{S}$. Then the generating-function conjecture.

$$\sum_{n\geq 0}|\operatorname{\mathsf{VHC}}(\operatorname{Av}_n(132,3241))|x^n=\sum_{n\geq 0}|\operatorname{\mathsf{VHC}}(\operatorname{Av}_n(231,2143))|x^n=\frac{1+x^2-\sqrt{1-4x+2x^2+x^4}}{2x}.$$

This conjecture proposes equal generating functions for valid hook configurations in two permutation-avoidance classes defined by one length-$3$ and one length-$4$ pattern. The paper presents it as a direction for future enumeration, and no resolution is supplied here.