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

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be reduced termwise modulo $2$. Modulo-2 periodicity conjecture. The Hankel transform of the sequence with generating function $1-x+x^2/r(x^2)$, taken modulo $2$, yields the periodic sequence with period

$$\overline{1,0,1,1,1,1,0,1}.$$

The conjecture follows observed initial terms in the paper's comparison of Rueppel and Catalan analogs; no proof or resolution is supplied.