The characteristic-polynomial conjecture for cross section lattices

Let $\Lambda^*$ be the cross section lattice of a $J$-irreducible monoid $M$, and let $J_0$ and $n$ have the meanings established in the preceding theorem. Characteristic-polynomial conjecture. Then

$$p(x,\Lambda^*)=x^{|J_0|}(x-1)^{n-|J_0|}.$$

The immediately preceding theorem proves this formula for supersolvable cross section lattices in the stated classical types. The conjecture proposes the same characteristic polynomial for cross section lattices of arbitrary $J$-irreducible monoids, but the supplied text gives no resolution.