Nederhof's multiple-context-freeness conjecture for free abelian groups

Let $W(G)$ denote the word problem of a group $G$ with respect to a finite symmetric generating set. In particular, $W(\mathbb{Z}^n)$ is the set of words over a finite generating set of $\mathbb{Z}^n$ that represent the identity element. Nederhof's conjecture. For any $n$, the language $W(\mathbb{Z}^n)$ is $n$-multiple context-free. This would sharpen the paper's bound of $(8\left[\frac{n+1}{2}\right]-2)$ and recover the known result that $W(\mathbb{Z}^2)$ is 2-multiple context-free; the proposed improvement remains open.