Minuscule or cominuscule periodicity conjecture for quantum twist automorphisms

Assume that the Weyl group is of finite type. Let $I$ be its index set, let $I_{\mathrm{MC}}$ consist of the minuscule or cominuscule indices, and let $x_t$ be the quantum twist element associated with $t\in I$. Write $\PD(x_t)$ for its periodicity.

Minuscule or cominuscule periodicity conjecture. Let $t\in I$. Then $\PD(x_t)$ is finite if and only if $t\in I_{\mathrm{MC}}$. Moreover, if $t\in I_{\mathrm{MC}}$, then the following hold:

• If the Weyl group is of type $B_n$ or $C_n$, then $\PD(x_1)=2$, while

$$\PD(x_n)=\begin{cases}2&\text{if }n=2,\\4&\text{otherwise.}\end{cases}$$

• If the Weyl group is of type $D_n$, then $\PD(x_1)=2$, while for $t\in\{n-1,n\}$,

$$\PD(x_t)=\begin{cases}2&\text{if }n=4,\\8&\text{if }n\text{ is odd},\\4&\text{otherwise.}\end{cases}$$

• If the Weyl group is of type $E_6$ and $t\in\{1,6\}$, then $\PD(x_t)=6$.
• If the Weyl group is of type $E_7$ and $t=7$, then $\PD(x_t)=4$.

The values are motivated by computational data. They have been verified for types $B_n$, $C_n$, $D_n$, $E_6$, and $E_7$ of rank at most $10$; outside $I_{\mathrm{MC}}$, computations found $\PD(x_t)>1000$.