The magic-number conjecture for strong cyclotomic Delone sets

Let $n\ge 3$, let $N=[n,2]$, and let $m_n$ denote the minimum number of pairwise nonparallel $\varLambda$-directions whose X-rays determine every convex subset of a strong $n$-cyclotomic Delone set $\varLambda$. Magic-number conjecture. With the exception of $m_4=7$, the trivial lower bound $m_n=N+1$ is the magic number for strong $n$-cyclotomic Delone sets for all $n\ge 3$. The exception reflects the affine equivalence of the triangular and square lattice cases, for which six lattice directions can contain a $U$-polygon.