Let $K$ be a compact Hausdorff space and $\mu$ be a Borel probability measure on $K$. Let $\tau=\{A_{i}\}_{i=1}^{n}$ be a partition of $K$ into finitely many Borel sets $A_{i}$ of positive measure. Define the operator $P_{\tau}:C(K)\rightarrow L_{p}(\mu)$ by $P_{\tau}(f)=\sum\limits_{i=1}^{n}\frac{\int_{A_{i}}fd\mu}{\mu(A_{i})}\chi_{A_{i}}$ for all $f\in C(K)$. 

Question. Is the $p$-nuclear norm of $P_{\tau}$ equal to 1?