The lcm-maximal piece conjecture for coefficient periods

For a piece $\mathbb P$ with move set $\mathbf{M}$, write

$$\hat d_r:=\max(|c_r|,|d_r|).$$

For a positive integer $\lambda$, let $\mathbb P^{\operatorname{lcm}}_\lambda$ be the lcm-maximal piece whose move set consists of all moves $(c,d)$ satisfying $\hat d\mid\lambda$. Lcm-maximal piece conjecture. Among all pieces satisfying

$$\operatorname{lcm}\{\hat d_r:m_r\in\mathbf{M}\}=\lambda,$$

$\mathbb P^{\operatorname{lcm}}_\lambda$ maximizes the period of every coefficient $\gamma_i$, for fixed $q$.

This conjecture is motivated by the proposed period bounds for $\gamma_3$ and $\gamma_4$; the paper gives no proof or general resolution.