$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra.

Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix of $L(P)$ which is defined as $-C^{-1} C^T$ when $C$ is the matrix with entries $c_{i,j}=1$ when $i \leq j$ and $c_{i,j}=0$ else for $i,j \in L$.

Question: For which types is the matrix $\Co_L$ periodic? That is when do we have $\Co_L^t=\operatorname{id}$ for some $t \geq 1$? What is the period (the smallest such $t$) in case this is true for a given type?

The quesiton has a positive answer for all simple Lie algebras of rank at most 5. So maybe it is always periodic?

For $A_n$ it is $2n+4$ for $n=2,3,4,5,6$ but for $A_1$ it is 3.

For $D_4$ it is 14 and for $D_5$ it is 18 and for $D_6$ it is 22.

For $B_2$ it is 10 and for $B_3$ it is 7 and for $B_4$ it is 18 and for $B_5$ it is 11 and for $B_6$ it is 26.

For $G_2$ it is 14 and for $F_4$ it is 26.