Explicit Hankel transform conjecture for the cubic Catalan family

Let $c(x)$ denote the Catalan generating function and define a sequence by

$$F_\alpha(x)=\frac{1-x-x^2-\alpha x^3}{1-x-x^2-x^3}c\left(\frac{x^3(1-x-x^2-\alpha x^3)}{(1-x-x^2-x^3)^2}\right).$$

Define

$$A_n(\alpha)=[x^n]\frac{1+(\alpha-2)x^2-(\alpha-2)x^3+(4\alpha-5)x^5-(\alpha-1)(\alpha-2)x^8}{1+3x^3-(\alpha-2)x^6},$$ $$B_n=[x^n]\frac{x(1-x+2x^2-2x^3+3x^4-3x^5+x^6)}{(1-x)^2(1-x^3)}.$$

Cubic-family Hankel-transform conjecture. The Hankel transform of the sequence with generating function $F_\alpha(x)$ is $h_n=A_n(\alpha)(2-\alpha)^{B_n}$. The formula is given as a further conjectural pattern after the quadratic-family examples.