Kato-Kuzumaki's properties for function fields over higher local fields

Let $k$ be a $d$-local field such that the corresponding $1$-local field $k^{(d-1)}$ is a $p$-adic field and $C$ a curve over $k$. Let $K$ be the function field of $C$. We prove that for each $n,m\in \mathbf{N}$, and hypersurface $Z$ of ${\mathbf{P}}_K^n$ with degree $m$ such that $m^{d+1}\le n$, the $(d+1)$-th Milnor $\mathrm{K}$-theory group is generated by the images norms of finite extension $L$ of $K$ such that $Z$ admits an $L$-point. Let $j\in \{1,\cdots ,d\}$. When $C$ admits a point in an extension $l\mathrm{/}k$ that is not $i$-ramified for every $i\in \{1,\cdots ,d-j\}$ we generalise this result to hypersurfaces $Z$ of ${\mathbf{P}}_K^n$ with degree $m$ such that $m^{j+1}\le n$. \par In order to prove these results we give a description of the Tate-Shafarevich group ${\text{Sha}}^{d+2}(K,\mathbf{Q}\mathrm{/}\mathbf{Z}(d+1))$ in terms of the combinatorics of the special fibre of certain models of the curve $C$.