Flat hypercomplex nilmanifolds are $\mathbb H$-solvable

Let $\mathbb H$ be a quaternion algebra generated by $I,J$ and $K$. We say that a hypercomplex nilpotent Lie algebra $\mathfrak g$ is $\mathbb H$-solvable if there exists a sequence of $\mathbb H$-invariant subalgebras containing $\mathfrak g_{i+1}=[\mathfrak g_i,\mathfrak g_i]$,
$$\mathfrak g=\mathfrak g_0\supset\mathfrak g_1^{\mathbb H}\supset\mathfrak g_2^{\mathbb H}\supset\cdots\supset\mathfrak g_{k-1}^{\mathbb H}\supset\mathfrak g_k^{\mathbb H}=0,$$
such that $[\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}]\subset\mathfrak g^{\mathbb H}_{i+1}$ and $\mathfrak g_{i+1}^{\mathbb H}=\mathbb H[\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}]$. Let $N=\Gamma\setminus G$ be a hypercomplex nilmanifold with the flat Obata connection and $\mathfrak g=\operatorname{Lie}(G)$. We prove that the Lie algebra $\mathfrak g$ is $\mathbb H$-solvable.