Existence of cyclotomic aperiodic substitution tilings with dense tile orientations

For each integer $n\geq 2$, consider cyclotomic aperiodic substitution tilings (CASTs) with dense tile orientations (DTO), finite local complexity (FLC), the minimal inflation multiplier specified in the source's table, and individual dihedral symmetry $D_{2n}$. Existence conjecture. Such CASTs exist for all $n\geq 2$.

This extends the explicitly constructed cases $n\in\{2,3,4,5,6,7\}$ and asserts that the apparent restriction to those cases is not intrinsic. The supplied text gives no proof or resolution, so the status remains open.