Distributive lattices with periodic Coxeter matrix

Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i leq j$ and $u_{i,j}=0$ else.
Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ is the Coxeter matrix of $L$.

Question: For which $L$ is the Coxeter matrix periodic, that is $C_L^n$ is the identity matrix for some $n$?

Call such a lattice (or its incidence algebra) periodic. For example any divisor lattice (products of chains) is periodic.
For distributive lattices on $r geq 2$ points, the number of periodic distributive lattices starts with 1,1,2,3,5,7,11,8,15 for $r leq 10$ and oeis finds this : .

Question: Is this still true for $r=11$?

Algebras with periodic coxeter matrix are studied in representation theory of quivers, see for example and several other articles.

One might also ask what the period is for a given lattice. For example for the Boolean lattice of $n$ points it is 3 if n is odd and 6 if n is even.
Recall that the free distributive lattice is the lattice of order ideals of the Boolean lattice.

Question: For which $n$ is the free distributive lattice on $n$ points periodic and what is the period in case it is? It is periodic with period 4 for $n=1$, period 10 for $n=2$ and with period 6 for $n=3$ and with period 42 for $n=4$.

Some tests suggest that the lattice of order ideals of a divisor lattice is also periodic.

MathOverflow Asked on January 27, 2021

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