Somos 4 conjecture for a quadratic $A$-matrix

Let $f(x)$ be determined by the quadratic equation

$$\frac{f(x)}{x}=1+af(x)+bf(x)^2+x\bigl(1+cf(x)+df(x)^2\bigr),$$

and let the Hankel transform be formed from the coefficient sequence of $f(x)/x$. A sequence is an $(\alpha,\beta)$ Somos $4$ sequence when

$$s_n s_{n-4}=\alpha s_{n-1}s_{n-3}+\beta s_{n-2}^2.$$

Somos 4 conjecture. The Hankel transform of the expansion of $f(x)/x$ is a

$$\left((b+ab+d)^2,\ b^4-b^3(2+3a+a^2-2c)+b(a+a^2-ac-2d)d+(1+a-c)d^2-b^2(c+ac-c^2+2d+3ad)\right)$$

Somos $4$ sequence.

This conjecture proposes a uniform Somos $4$ recurrence for the family of quadratic $A$-matrices considered in the paper; the supplied text does not state whether it has been proved or disproved.