The A005811 Hankel transform for $1-x+x^2/(1+x^2r(x^2))$

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be the sequence of determinants of successive Hankel matrices. A005811 Hankel-transform conjecture. The Hankel transform of the sequence with generating function

$$1-x+\frac{x^2}{1+x^2r(x^2)}$$

is given by OEIS sequence A005811 evaluated at $n-1$. The paper supports this with initial terms and a parameterized calculation, but supplies no proof or resolution.