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

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 $x+1/r(x^2)$ is a signed version of OEIS sequence A005811 evaluated at $n+1$. The source identifies A005811 as counting the number of runs in the binary expansion, equivalently the number of $1$'s in the Gray code, and gives numerical evidence but no proof or resolution.