Transitive-element conjecture for the property $\mathcal{P}(H,G)$ in binary wreath products

Let $W_n$ be the ambient binary wreath-product group acting on $\{1,\dots,2^n\}$. For $n\geq 1$, let $H\leq W_n$ be a subgroup containing a permutation $\sigma\in W_n$ that acts transitively on $\{1,\dots,2^n\}$. Let $G\leq W_n$ be any subgroup. Transitive-element conjecture. Under these hypotheses, $\mathcal{P}(H,G)$ holds. The conjecture is based on extensive Magma computations and proposes a sufficient transitivity condition for the property $\mathcal{P}(H,G)$; the supplied material gives no proof or resolution.