Domokos and Zubor's dimension conjecture for maximal commutative subalgebras of Grassmann algebras

Let $F$ be a field of characteristic different from two, and let $G(n)$ be the Grassmann algebra

$$G(n)=F[x_1,\ldots,x_n]/\langle x_ix_j+x_jx_i\mid 1\leq i,j\leq n\rangle_F.$$

A maximal commutative subalgebra is a commutative subalgebra maximal with respect to inclusion. If $n=4k+1$ and $A$ is a maximal commutative subalgebra of $G(n)$, then Domokos and Zubor's dimension conjecture.

$$\dim(A)\geq 3\cdot 2^{n-2}.$$

For even $n$, every maximal commutative subalgebra has dimension $3\cdot 2^{n-2}$, whereas the odd-dimensional case has a less clear structure. The conjecture is refuted by the constructions described in the paper, which produce counterexamples for $n=4k+1$ with $k\geq 4$.