Three regular polytope conjecture for multicomplex Mandelbrot sets

Let $n\geq3$. The multicomplex Mandelbrot set ${\mathcal{M}_{n}}$ is considered through its principal and idempotent $n$-dimensional slices, including slices associated with regular convex polytopes. Three regular polytope conjecture. The set ${\mathcal{M}_{n}}$ contains exactly three regular convex $n$-polytopes among all possible principal or idempotent $n$-dimensional slices. This conjecture is motivated by the tesseract found in an idempotent 4D slice and the expected regular four-dimensional cross-polytope in another slice, together with the three Platonic solids identified in the tricomplex case.