Reduction conjecture for Hecke cycles below the cohomological dimension

Let $\xi$ and $\xi'$ be the cycles associated with Algorithm~, and suppose $n\leq4$ and $\|\xi\|>1$. For each modular symbol ${\mathbf v}$ in step A of the algorithm, choose $w({\mathbf v})$ to be a good candidate, meaning a candidate minimizing the maximum norm of the resulting modular symbols. Hecke-cycle reduction conjecture. If every $w({\mathbf v})$ is chosen using either the Voronoi candidate conjecture or the LLL candidate conjecture, then

$$\|\xi'\|<\|\xi\|.$$

The claim would establish termination and practical effectiveness of the proposed Hecke-eigenvalue algorithm in the dimensions relevant to the paper. The authors report that the inequality held in computer experiments for $n\leq4$, but do not prove it.