Ragunathan–Van Tuyl conjecture on the Weak Lefschetz Property of van der Waerden rings

Let $n>k\geq 1$ be integers, let ${\tt vdw}(n,k)$ be the simplicial complex whose facets are the arithmetic progressions of length $k$ in $\{1,\ldots,n\}$, and set

$$A(n,k)=K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)}+\langle x_1^2,\ldots,x_n^2\rangle),$$

where $I_{{\tt vdw}(n,k)}$ is the associated Stanley–Reisner ideal. The ring $A(n,k)$ has the Weak Lefschetz Property (WLP) if multiplication by a general linear form has maximal rank between every pair of consecutive graded pieces.

Ragunathan–Van Tuyl's conjecture. Fix an integer $k\geq 3$. If $k$ is odd, then there exists an $m$ such that $A(n,k)$ fails to have the WLP for every $n\geq m$. If $k$ is even, then $A(n,k)$ has the WLP for all $n>k$.

The conjecture extends the cases established in the paper for $k=1,2,3$ and is based on computations for $1\leq k<n\leq20$. It predicts eventual failure in every odd column with $k\geq3$, while asserting the WLP for all $n>k$ in every even column.