Periodicity in minuscule and cominuscule cases beyond type A

Prove that the periodicity PD(x_t) of the crystal-theoretic quantum twist automorphism is finite if and only if t is a minuscule or cominuscule index, and establish the stated values for types B_n, C_n, D_n, E_6, and E_7.

Background

The paper defines PD(w) as the least positive power of the crystal-theoretic twist that returns every localized-crystal element up to frozen variables. Type A is handled theoretically earlier in the paper, while computations suggest a sharp distinction in the other finite classical and exceptional types: finite periodicity occurs precisely at minuscule or cominuscule indices. The conjectured numerical periods are based on SageMath computations, including cases up to rank 10, but are not proved.