Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$?

(i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-Scholze.

(ii) I know that $\pi_n THH(\mathbb{Z}_p)$ is $\mathbb{Z}_p$ for $n=0$, $\mathbb{Z}_p/m$ for $n=2m-1 \geq 0$, and zero otherwise. I tried to write down the Tate spectral sequence for $\pi_* TP(\mathbb{Z}_p)$ but it seems very complicated. I think Tsalidis and Bokstedt-Madsen calculated the homotopy groups of $TP(\mathbb{Z}_p)/p$ but I don't know how to get $\pi_* TP(\mathbb{Z}_p)$ from this.

(iii) A related question is what is the Breuil-Kisin twisted prismatic cohomology $\Delta_{\mathbb{Z}_p}\{i\}$?