This is a local result so we can work with the geodesic ball $B_{r}(p)$ for any $p\in P$ such that $\overline{B_{r}(p)}\subset\subset Q$ and $\overline{B_{r}(p)}\cap P\subset\subset P$. We will show that the $4$-current $dG_{P}$ is a harmonic in $B_{r}(p)$ in the distributional sense. In fact, for any test form $\chi\in\Omega^{m-4}_{0}(B_{r}(p))$ we have

(3.44)

$$(\Delta(d(G_{P})),\chi)=(d\Delta G_{P},\chi)=(\Delta G_{P},d\chi)=(2\pi\delta_{P},d\chi)=2\pi\int_{P}d\chi=2\pi\int_{\partial B_{r}(p)\cap P}\chi=0.$$

Therefore, $d(G_{P})$ is a harmonic $4$-current in $B_{r}(p)$ and hence it is smooth in $B_{r}(p)$. ∎