Somos 4 conjecture for the elliptic-curve family sequence

Let $b_n$ be the sequence whose generating function is

$$g(x)=\frac{1+3t+t^2-x}{1+2(t+2)x}c\left(\frac{x^2(1+3t+t^2-x)}{(1+2(t+2)x)^2}\right),$$

arising from the elliptic-curve family $E_t:y^2+4xy+y=x^3+(t-1)x^2+tx$. A $(\alpha,\beta)$ Somos $4$ sequence satisfies

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

Elliptic-curve family Somos 4 conjecture. The Hankel transform of $b_n$ is a

$$\left((2t^3+10t^2+14t+5)^2,-3t^8-40t^7-222t^6-666t^5-1173t^4-1230t^3-740t^2-29(8t+1)\right)$$

Somos $4$ sequence. The conjecture links this family of elliptic curves to a parameterized Somos $4$ recurrence.