Somos 4 conjecture for the transformed elliptic-curve sequence

Let $t$ be a parameter and let $a_n$ be the sequence whose generating function is

$$\frac{1+(2t+5)x}{1+2(t+3)x+(t+2)(t+3)x^2}c\left(\frac{x^3(1+(2t+5)x)}{(1+2(t+3)x+(t+2)(t+3)x^2)^2}\right).$$

Transformed elliptic-curve Somos 4 conjecture. The Hankel transform of $a_n$ is a $(1,t^2+3t+1)$ Somos $4$ sequence. This sequence is obtained from the preceding elliptic-curve construction by a generating-function transformation.