The SLP conjecture for nonequipointed linear spaces

Let $(P,L)$ be a linear space with $d=\#P$, and let $A$ be its associated ring. Set $n$ and $m$ to be the minimum and maximum, respectively, size of a line in $L$. Suppose $2\leq n<m$. Based on experimental evidence, the SLP conjecture. The ring $A$ has the strong Lefschetz property if and only if $\operatorname{char} K=0$ or $\operatorname{char} K>m$, and one of the following conditions holds:

1. $3\leq m\leq4$;
2. $m=5$ and every point is on a line of size at least $4$;
3. $m\geq6$ is even and $n+1=m$.

The preceding result characterizes the weak Lefschetz property for nonequipointed linear spaces; this conjecture proposes the corresponding characterization for the strong Lefschetz property, but the paper provides only experimental evidence.