Conjecture identifying a Hankel transform with the Robbins numbers

Let $\tilde{A}(x)$ be the generating function constructed from the revert transform of the Fibonacci numbers as in the source, and consider the generating function

$$\tilde{A}(ix)=\cfrac{1}{1+\cfrac{x^2}{1-\cfrac{x^2}{1+ix-\cfrac{x^2}{1+ix-\cfrac{x^2}{1+ix-\cdots}}}}}.$$

Its revert transform begins

$$1,0,1,0,3,-i,12,-11i,51,-89i,204,-628i,646,\ldots.$$

Robbins-number conjecture. The Hankel transform of this revert transform is the Robbins number sequence

$$1,1,2,7,42,429,7436,\ldots.$$

The claim is presented as a conjecture connecting a continued-fraction generating function and the Robbins numbers; the source gives no resolution.