Higher-spatial-regularity conjecture for the Augmented formula

Let $n\geq 2$ with $n\in\mathbb{N}$. Let $\mathbb{H}^n_\kappa$ be the two-dimensional vectorial field equipped with the weighted norm specified below. For even $n$, define

$$\begin{aligned} \lVert u\rVert_{\mathbb{H}^{n}}^2:= {}&\lVert u_1\rVert_{L^2_r}^2+\lVert\partial_r u_1\rVert_{L^2_r}^2+\lVert {\rm d}_r\partial_r u_1\rVert_{L^2_r}^2+\cdots+\lVert(\partial_r{\rm d}_r)^{n/2}u_1\rVert_{L^2_r}^2\\ &+\lVert u_2\rVert_{L^2}^2+\lVert{\rm d}_r u_2\rVert_{L^2_r}^2+\lVert\partial_r{\rm d}_r u_2\rVert_{L^2_r}^2+\cdots+\lVert({\rm d}_r\partial_r)^{n/2}u_2\rVert_{L^2_r}^2+\kappa^2\lVert({\rm d}_r\partial_r)^{n/2}{\rm d}_r u_2\rVert_{L^2_r}^2. \end{aligned}$$

For odd $n$, define

$$\begin{aligned} \lVert u\rVert_{\mathbb{H}^{n}}^2:= {}&\lVert u_1\rVert_{L^2_r}^2+\lVert\partial_r u_1\rVert_{L^2_r}^2+\lVert {\rm d}_r\partial_r u_1\rVert_{L^2_r}^2+\cdots+\lVert({\rm d}_r\partial_r)^{(n-1)/2}{\rm d}_r u_1\rVert_{L^2_r}^2\\ &+\lVert u_2\rVert_{L^2}^2+\lVert{\rm d}_r u_2\rVert_{L^2_r}^2+\lVert\partial_r{\rm d}_r u_2\rVert_{L^2_r}^2+\cdots+\lVert{\rm d}_r(\partial_r{\rm d}_r)^{(n-1)/2}u_2\rVert_{L^2_r}^2+\kappa^2\lVert(\partial_r{\rm d}_r)^{(n+1)/2}u_2\rVert_{L^2_r}^2. \end{aligned}$$

Higher-spatial-regularity conjecture. The Augmented formula is well-posed in $\mathscr{C}^k\left([0,T),\mathbb{H}^n\times\mathbb{R}^4\right)$, with the corresponding weighted norm above according to the parity of $n$. The conjecture is motivated by applying the indicated alternating radial differential operators to the Augmented formula. The supplied passage gives no result establishing or refuting this higher-regularity claim.