Somos 4 conjecture for generalized Schröder Hankel transforms

Let $a_n(p,s,t)$ be the generalized Schröder numbers and let $h_n(p,s,t)$ be their Hankel transform. A $(\alpha,\beta)$ Somos $4$ sequence satisfies

$$e_n=\frac{\alpha e_{n-1}e_{n-3}+\beta e_{n-2}^2}{e_{n-4}}.$$

Generalized Schröder Somos 4 conjecture. The Hankel transform $h_n(p,s,t)$ is a $((pst)^2,(pt)^3(pt-s^2))$ Somos $4$ sequence. This would connect the generalized Schröder Hankel transform with elliptic-curve phenomena associated with Somos $4$ recurrences.