A hyperbolic distance identity for intersections of lines in the Poincaré disk

Let $a,b,c,d$ be four complex points on the unit circle $S^1$ in this order, with the Euclidean lines $L(a,b)$ and $L(c,d)$ nonparallel. Let $h$ be an arbitrary point on the Euclidean segment $[b,c]$. Define

$$g={\rm LIS}[a,b,c,d],\qquad j={\rm LIS}[g,h,a,c],\qquad k={\rm LIS}[g,h,b,d],\qquad l={\rm LIS}[g,h,a,d].$$

Here ${\rm LIS}$ denotes the line-intersection construction used in the paper, and the special case $j=k$ is allowed. Hyperbolic distance identity. The hyperbolic distances in the Poincaré disk satisfy

$$\rho_{\mathbb{B}^2}(h,j)=\rho_{\mathbb{B}^2}(k,l).$$

This is a metric relation for the line-intersection construction associated with four cyclic points. The supplied text does not establish whether the statement is proved or remains open.