The Hankel-transform relation for Rueppel complements and first differences

Let $r_n$ be the Rueppel sequence. Let $H_n$ be the Hankel transform of the sequence $1-r_n$, and let $h_n$ be the Hankel transform of the first-difference sequence $r_{n+1}-r_n$. Rueppel complement-difference conjecture. Then

$$|h_n|=\sqrt{|H_{n+1}|-|H_n|}.$$

The paper notes analogous relations for the Catalan and Motzkin numbers, but gives no proof or resolution for the Rueppel relation.