Bleher–Chinburg–Greenberg–Kakde–Pappas–Sharifi–Taylor pseudo-nullity conjecture for a finite character deformation

Let $K$ be an imaginary quadratic field in which $p$ splits, let $\chi:\operatorname{Gal}(\overline{\mathbb{Q}}/K)\to\mathbb{Z}_p^\times$ be a finite continuous character, and let $\widetilde{\Gamma}_K$ and $\widetilde{\kappa}$ be as above. Consider the deformation $\chi\widetilde{\kappa}^{-1}$ and its associated Tate–Shafarevich group. The pseudo-nullity conjecture for the finite character deformation. The $\mathbb{Z}_p[[\widetilde{\Gamma}_K]]$-module

$$\Sha^1\left(K,\mathcal{D}_{\chi\widetilde{\kappa}^{-1}}\right)^\vee$$

is pseudo-null. The conjecture is described as a modification of the preceding pseudo-nullity conjecture and as motivation for a main result of Bleher and collaborators; the source does not state that it has been resolved.