Higher-time regularity conjecture for the Augmented formula

Let $k\in\mathbb{N}$, and let $T_{\varepsilon,\kappa,R}$ denote the time scale introduced for the Augmented formula. The solution takes values in the product of the Sobolev space $\mathbb{H}^1$ and $\mathbb{R}^4$. Higher-time regularity conjecture. For any $k\in\mathbb{N}$, the Augmented formula is well-posed in

$$\mathscr{C}^k\left([0,cT_{\varepsilon,\kappa,R}),\mathbb{H}^1\times\mathbb{R}^4\right).$$

The claim is motivated by applying $\partial_t^k$ to the Augmented formula and seeking to use the same semilinear scheme. The supplied passage gives no result establishing or refuting this conjecture.