Kisin's connectedness conjecture for Kisin varieties

Let $K$ be a finite extension of $\Q_p$ with $p>2$, let $\mathbb F$ be a finite field, and let

$$\bar\rho:\Gamma_K\rightarrow\operatorname{GL}_n(\mathbb F)$$

be an $n$-dimensional continuous representation of the absolute Galois group $\Gamma_K$. Let $C_{\mu}(\bar\rho)$ be the Kisin variety parametrizing finite flat group schemes over $\mathcal O_K$ with generic fiber $\bar\rho$ and determinant condition determined by $\mu$. Kisin's connectedness conjecture. If $\bar\rho$ is absolutely irreducible, then $C_{\mu}(\bar\rho)$ is connected. Kisin's conjecture is known in dimension two and in particular when the ramification index satisfies $e(K/\mathbb Q_p)<p-1$, while the connectedness question for general absolutely irreducible representations is the subject of this paper.