Homogeneous realization conjecture for Grassmann algebra gradings

Let $E$ be the Grassmann algebra, and let a homogeneous $\mathbb{Z}_{2}$-grading of $E$ mean a grading in which the canonical Grassmann generators are homogeneous. Two gradings are $\mathbb{Z}_{2}$-graded isomorphic if there is an algebra isomorphism preserving their $\mathbb{Z}_{2}$-degree decompositions.

Homogeneous realization conjecture. Every $\mathbb{Z}_{2}$-grading of $E$ is $\mathbb{Z}_{2}$-graded isomorphic to some homogeneous $\mathbb{Z}_{2}$-grading of $E$.

The preceding theorem shows this for the automorphisms constructed by Methods A, B, and C, whose associated superalgebras are isomorphic to one arising from an automorphism of type 1. The conjecture, first stated in the cited work, remains open.