Polynomial-move conjecture for enlarging regular polygons in the grasshopper game

Let $N>6$ and consider a configuration of grasshoppers at the vertices of a regular $N$-gon, with legal moves given by the grasshopper-jumping game described in the paper. Polynomial-move conjecture. Every regular $N$-gon can be transformed to a similar but larger configuration in polynomially many moves in $N$. The main theorem establishes that a finite sequence of moves always exists, while the conjecture asks for a polynomial bound; the explicit bound obtained from the proof is unreasonably large and may be exponential.