Gilbert's conjecture on Real Hardy spaces for Clifford modules

Let $\mathfrak{H}$ be a Clifford module. The claim concerns an element $\eta\in\mathbb{R}^n$ with $\eta^2=-1$, a subspace $\mathfrak{H}_0\subseteq\mathfrak{H}$, the Cauchy integral operator $C$, the boundary operator $\mathcal B$, and an operator $\mathcal R$ equal to twice the orthogonal projection onto $\mathfrak{H}_0$.

Gilbert's conjecture. There exist $\eta$ and $\mathfrak{H}_0$ such that

$$\mathfrak{H}=\mathfrak{H}_0\oplus\eta\mathfrak{H}_0,$$

$C$ is an isomorphism

$$C:L^p(\mathbb{R}^{n-1},\mathfrak{H}_0)\longrightarrow H^p(\mathbb{R}_{+}^n,\mathfrak{H})$$

for every $p>1$, and $\mathcal R\mathcal B$ maps $H^p(\mathbb{R}_{+}^n,\mathfrak{H})$ continuously onto $L^p(\mathbb{R}^{n-1},\mathfrak{H}_0)$ for every $p>1$.

The conjecture proposes a Real $H^p$ theory for Hardy spaces valued in Clifford modules and is presented as a formulation of a question attributed to Gilbert. The supplied text gives no evidence that it has been resolved.