Böckle–Juschka irreducible-component conjecture for framed deformation spaces

Let $p$ be a prime, let $n$ be a positive integer, let $K$ be a finite extension of $cQ_p$, and let $\overline\rho:G_K\to\operatorname{GL}_n(k_L)$ be a continuous representation. Let $\mathcal{X}^{\square}_{\overline\rho}$ be the rigid generic fiber of the framed deformation ring of $\overline\rho$, and let

$$d:\mathcal{X}^{\square}_{\overline\rho}\to\mathcal{X}^{\square}_{\det\overline\rho}$$

be the map induced by sending a deformation to its determinant. Böckle–Juschka's conjecture. The map $d$ induces a bijection between the irreducible components of the two spaces, and the irreducible and connected components of $\mathcal{X}^{\square}_{\overline\rho}$ coincide. This conjecture concerns the geometry of unrestricted framed local deformation spaces; the source reports that it is resolved in several special cases, including $n=1$, certain two-dimensional cases, and cases with a vanishing adjoint cohomology group, while the general statement is left open.