Termination conjecture for the pentagon contact representation algorithm

Let $G$ be an inner triangulation of a $5$-gon, and let $F$ be an initial five color forest of $G$. The algorithm described above produces iterates for these data.

Termination conjecture. The algorithm terminates with a non-negative solution for every graph $G$ which is an inner triangulation of a $5$-gon, and for every initial five color forest $F$ of $G$.

The conjecture asserts termination and success of the algorithm for all admissible inputs. The paper reports that two independent implementations and experiments have always been successful, but the authors cannot prove that the iterations make progress and note that the algorithm might otherwise cycle forever.