Let $p\in \mathbb{N}$, and define $\mathrm{Lip}_p$ be the collection of functions from $\mathbb{R}^d$ to itself, with $p-1$ first derivatives bounded and whose $p^{\mathrm{th}}$ derivative is Lipschitz. This can be made into a Banach space using the norm
$$
\|f\| \triangleq
\max_{| i | \leq p-1} \sup_{x\in\mathbb{R}^d} \left \|(D^if) (x) \right \|_{\mathbb{R}^d}
+
\mathrm{Lip}(f^{(p)}),
$$
where $\mathrm{Lip}(h)$ assigns to a Lipschitz function $h$ its minimal Lipschitz constant. *(This definition is analogous to that of a Hölder space)*

In this article, it was shown that if $p=0$, then the predual of $\mathrm{Lip}_0$ is uniquely defined. Is this still the case for positive integers $p$?