2 problems

Periodicity conjecture. The minimal periods of the sequences $a^{r_1}(X_n)$ and $a^{r_0}(X_n)$ are equal.

Booher–Cais conjecture. For each $r\geq 1$, there exist an integer $m_r$ and functions $b_r,\nu_r,\lambda_r:\mathbf Z/m_r\mathbf Z\to\mathbf Q$ such that, for $n\gg0$,