Minimal-dimension conjecture for the constructed Grassmann-algebra subalgebra

Let $G(4k+9)$ be the Grassmann algebra, let $G_0$ denote its even component, and let $M_{\mathcal{C}}$ be the subspace associated with the constructed maximal commutative algebraic system $\mathcal{C}$. Grassmann-algebra minimality conjecture. Is the dimension of $G_0\oplus M_{\mathcal{C}}$ minimal among all maximal commutative subalgebras of $G(4k+9)$ for $k\geq 2$? The construction yields maximal commutative subalgebras and, in the verified range, dimensions below $3\cdot 2^{n-2}$; the conjecture asks whether this particular dimension is globally minimal.