$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

For $n\ge2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \to \mathbb{R}^{n\times n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}$ and generalized vanishing tangential trace $P\,\tau_l=0$ on $\partial \Omega$, denoting by $\{\tau_l\}_{l=1,\ldots, n-1}$ a moving tangent frame on $\partial\Omega$, more precisely we have: $$\| P \|_{L^p(\Omega,\mathbb{R}^{n\times n})}\leq c\,(\| \operatorname{sym} P\|_{L^p(\Omega,\mathbb{R}^{n \times n})} + \|\underline{\operatorname{Curl}} P \|_{L^p(\Omega,(\mathfrak{so}(n))^n)} ),$$ where the generalized $\underline{\operatorname{Curl}}$ is given by $(\underline{\operatorname{Curl}})_{ijk} :=\partial_i P_{kj}-\partial_j P_{ki}$ and $c=c(n,p,\Omega)>0$.