The pyramid conjecture for Gorenstein polytopes of degree d

Let $P$ be an $n$-dimensional Gorenstein polytope of degree $d$. A polytope is a pyramid if it is the lattice pyramid over another lattice polytope. Pyramid conjecture. Every $3d$-dimensional Gorenstein polytope $P$ of degree $d$ is a pyramid. Moreover, up to isomorphism, there exists a unique $(3d-1)$-dimensional Gorenstein polytope $\Theta_d$ of degree $d$ that is not a pyramid. The classification verifies this conjecture when $d=2$, but it remains open for $d\geq 3$.