The periodic Hankel transform of $1-x/r(x^2)$

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform of a sequence be the sequence of determinants of its successive Hankel matrices. Periodic Hankel-transform conjecture. The Hankel transform of the sequence with generating function $1-x/r(x^2)$ is the periodic sequence

$$1,-1,-1,0,1,-1,-1,0,1,-1,-1,0,\ldots.$$

The claim is based on the displayed initial Hankel-transform values and is part of the paper's study of Rueppel analogs of Catalan-related sequences; no proof or resolution is given.