Uniqueness of maximal-dimensional commutative subalgebras for n congruent to 3 modulo 4

Let $E$ be the Grassmann algebra and let $A_1,A_2\subseteq E$ be maximal-dimensional commutative subalgebras. Uniqueness conjecture. If $n=4k+3$, then $A_1\cong A_2$. This asks whether all maximal-dimensional commutative subalgebras are isomorphic in the case $n\equiv 3\pmod 4$; the supplied text gives no resolution.