The local crystalline-lift conjecture for Serre weights

Let $K/{\mathbb{Q}}_p$ be a finite totally ramified extension, and let ${\overline{\rho}}:G_K\to\operatorname{GL}_2({\overline{{\mathbb{F}}}_p})$ be a continuous representation. Let $a\in{\mathbb{Z}}^2_+$ be a Serre weight. For a lift $\lambda\in({\mathbb{Z}}^2_+)^{{\operatorname{Hom}}_{{\mathbb{Q}}_p}(K,{\overline{{\mathbb{Q}}}_p})}$ of $a$, suppose there is a continuous crystalline representation $\rho:G_K\to\operatorname{GL}_2({\overline{{\mathbb{Q}}}_p})$ lifting ${\overline{\rho}}$ and having Hodge type $\lambda$. The local crystalline-lift conjecture. Under these hypotheses, $a\in W^?({\overline{\rho}})$. This conjecture proposes that the existence of a crystalline lift with the prescribed Hodge type forces the corresponding predicted local Serre weight; together with the cited local theorem, it would essentially resolve the global Serre weight conjecture. Its resolution is not given in the source.