Local analytic Cauchy problem for the parabolic monogenic equation

Let $\mathcal U$ be the relevant parabolic domain, let $M(n-1,k,\mathbb R)$ be the subspace defined by

$$x_{11}=\ldots=x_{1k}=y_{12}=\ldots=y_{k-1,k}=0,$$

and let $\mathcal V$ be an open subset of $M(n-1,k,\mathbb R)$. Given a real analytic spinor $\psi$ in the variables $x_{\alpha i}$ with $\alpha\geq 2$ that converges on $\mathcal V$, the local Cauchy problem asks for a parabolic monogenic spinor $\Psi$ on a neighborhood of $\mathcal V$ in $\mathcal U$.

Local extension conjecture. There is a unique parabolic monogenic spinor $\Psi$ converging on a neighborhood of $\mathcal V$ in $\mathcal U$ such that

$$D\Psi=0$$

and whose restriction to $M(n-1,k,\mathbb R)$ coincides with $\psi$.

This asserts local existence and uniqueness for the analytic Cauchy problem associated with the parabolic monogenic equation, extending prescribed analytic spinor data from the indicated subspace. The supplied text does not state whether this claim has been proved or remains open.