Reflection-involution characterization conjecture for length-equivalent orthogeodesics

Let $\alpha$ and $\beta$ be orthogeodesics. They are length equivalent when they have equal lengths. A reflection involution is an involution reflection acting on the relevant hyperbolic surface. Reflection-involution characterization conjecture. The orthogeodesics $\alpha$ and $\beta$ are length equivalent if and only if there is a finite sequence of reflection involutions $r_1,r_2,\ldots,r_n$ such that

$$r_1 \circ r_2 \circ \cdots \circ r_n(\alpha)=\beta.$$

Equivalently, there is a finite sequence of r-ortho-isosceles-trapezoids connecting $\alpha$ and $\beta$. This is presented as a conjecture related to the length-equivalent closed-geodesic problem on hyperbolic surfaces; the source gives no evidence that it has been resolved.