The solvability conjecture for nilpotent hypercomplex Lie algebras

Let $(\mathfrak{g}, \{J_{\alpha}\})$ be a nilpotent hypercomplex Lie algebra. Here, $\mathbb{H}$-solvability means that the descending $\mathbb{H}$-central series defined by $\mathfrak{g}_0^\mathbb{H}=\mathfrak{g}$ and $\mathfrak{g}_k^\mathbb{H}=\mathbb{H}[\mathfrak{g}_{k-1}^\mathbb{H},\mathfrak{g}_{k-1}^\mathbb{H}]$ eventually reaches zero. Solvability conjecture. Every nilpotent hypercomplex Lie algebra $(\mathfrak{g}, \{J_{\alpha}\})$ is $\mathbb{H}$-solvable. This would establish that nilpotency forces the stronger $\mathbb{H}$-solvability property; the preceding discussion notes that the properness of $\mathfrak{g}_1^\mathbb{H}=\mathfrak{g}^1+J_1\mathfrak{g}^1+J_2\mathfrak{g}^1+J_3\mathfrak{g}^1$ is necessary, but does not resolve the conjecture.