The $\gamma_3$ period conjecture for rider pieces

Let $\mathbf{M}$ be the move set of a piece $\mathbb P$, with basic moves $(c_r,d_r)$, and write

$$\hat d_r:=\max(|c_r|,|d_r|),\qquad \Lambda:=\operatorname{lcm}\{\hat d_r:m_r\in\mathbf{M}\}.$$

Let $\gamma_3$ denote the third coefficient of the quasipolynomial $u_{\mathbb P}(q;n)$. The $\gamma_3$ period conjecture. The period of $\gamma_3$ is either $1$ or $\Lambda$.

The claim is supported in the paper by several examples, including two pieces, arbitrary numbers of one-move riders, three partial queens, and two partial nightriders, but no general proof is given.