The algorithmic termination conjecture for rational relation numbers

Let $s,r\in\mathbb{N}$ satisfy $s/r<4$. For each positive integer $M$, let $\textsc{RelNum}(s,r,M)$ and $\textsc{RelNumMin}(s,r,M)$ denote the two algorithms defined in the paper, with the latter using the coordinate-minimizing procedure.

Algorithmic termination conjecture. For all $s,r\in\mathbb{N}$ satisfying $s/r<4$, there exists $M>0$ such that

$$\textsc{RelNum}(s,r,M)=\texttt{True}\quad\text{or}\quad \textsc{RelNumMin}(s,r,M)=\texttt{True}.$$

If true, this would provide an algorithmic certificate for the Main Conjecture for the corresponding rational parameters. The paper states that this conjecture would imply the Main Conjecture, but does not establish it in general.