The maximal-move-range piece conjecture for coefficient periods

For a piece $\mathbb P$ with move set $\mathbf{M}$, define its move range by

$$\|\mathbb P\|:=\max_{(c_r,d_r)\in\mathbf{M}}\max(|c_r|,|d_r|).$$

For a positive integer $\lambda$, let $\mathbb P^{\max}_\lambda$ be the piece with move set

$$\mathbf{M}_\lambda:=\{(c,d): |c|,|d|\leq\lambda,\ \gcd(c,d)=1\}.$$

Maximal-move-range piece conjecture. Among all pieces with $\|\mathbb P\|\leq\lambda$, $\mathbb P^{\max}_\lambda$ maximizes the period of every coefficient $\gamma_i$, for fixed $q$.

This is proposed as a practical universal principle for bounding coefficient periods; the paper states it as a conjecture without supplying a proof or a general resolution.