. Do shortest paths walking between rational points of height $h$ ever properly cross themselves?Q

Explaining this question takes a bit of definitional exposition.

First, I copy definitions from an earlier question,
"Circles avoiding rational points of height ≤h."
A *rational point* in $\mathbb{R}^2$ is a point whose coordinates are both rational numbers.
A rational number $x= a/b$ in lowest terms
(i.e., gcd$(a,b)=1$) has *height*
$\max \lbrace |a|,|b| \rbrace$.
A *rational point
of height $h$* is a rational point whose coordinates are both of height $\le h$.

Now we take as our domain all the rational points in $(0,1)^2$,
and define a metric on the set of pairs $(a, h(a))$.
Let
$$d(a,b) = \|a-b\| \cdot \tfrac{1}{2} {\left(h(a) + h(b) \right)} \;:$$
the Euclidean distance between $a$ and $b$
times their average $h$-height,
i.e., the area of the "*fence*" under the $(a,h(a)){-}(b,h(b))$-path.

Now imagine seeking the shortest path between two points $a,b \in (0,1)^2$ under this metric. For example, here is one path from $(\frac{1}{8} , \frac{1}{6})$ to $(\frac{7}{8} , \frac{7}{8})$:

^{ $$ \left( \frac{1}{8} , \frac{1}{6} , 8 \right) \;, \left( \frac{1}{6} ,\frac{1}{2} , 6 \right) \;, \left( \frac{1}{3} , \frac{1}{2} , 3 \right) \;, \left( \frac{7}{8} , \frac{7}{8} , 8 \right). $$ }

^{ Heights of points shown circled. }

The path illustrated is shorter than the direct, one-step path from $(\frac{1}{8},\frac{1}{6})$ to $(\frac{7}{8},\frac{7}{8})$ (but it is not the shortest path).

Does a shortest $ab$ path ever properly cross itself, in the sense that the "fences" (properly) intersect?

(Of course adjacent fences share a vertical edge.)
Perhaps the answer is *Yes* but I don't see a counterexample.
Generally shortest paths do not properly cross,
which is one reason I think this could be interesting.