Lefschetz properties for codimension-four Artin Gorenstein rings with prescribed regularity

Let $A$ be an Artin Gorenstein ring with codimension $c$ and regularity $r$, and let a Betti table refer to the graded Betti table of a minimal free resolution of $A$. The codimension-four Lefschetz conjecture. For $c=4$ and $r=5$, there are 36 possible Betti tables; any such ring whose Betti table does not appear in Theorem 54 has the strong Lefschetz property, and hence the weak Lefschetz property. For $c=4$ and $r=6$, every such Artin Gorenstein ring has the weak Lefschetz property. These assertions are presented as computational evidence and are not accompanied by a resolution in the source.