Asymptotic expected-length conjecture for grand zigzag knight's paths

Let a grand zigzag knight's path be a path whose number of steps has an expected value determined by the corresponding family of paths ending on the $x$-axis at size $n$. Asymptotic expected-length conjecture. An asymptotic approximation for the expected number of steps of a grand zigzag knight's path ending on the $x$-axis of size $n$ is

$$\frac{1+\sqrt{5}}{2\sqrt{5}}\cdot n.$$

The even-size case is established in the preceding theorem, while the source presents the odd-size analogue as suggested by Mathematica calculations and does not provide a proof.