The equality of the finite groups $\mathcal{J}_n^{p,q}$ and $\mathcal{K}_n^{p,q}$

Let $p$ and $q$ be the parameters defining the groups $\mathcal{J}_n^{p,q}$ and $\mathcal{K}_n^{p,q}$ associated with the level-$n$ action on the binary tree. The preceding theorem establishes the inclusion $\mathcal{K}_n^{p,q}\subseteq\mathcal{J}_n^{p,q}$.

Equality conjecture. For all relevant $n$, $p$, and $q$, one has

$$\mathcal{J}_n^{p,q}=\mathcal{K}_n^{p,q}.$$

Computer calculations for $(p,q)=(1,2)$ and $(3,2)$ give matching group orders through $n=12$, motivating the conjectured reverse inclusion. The source provides no proof of the equality.