Local higher-weight crystalline lift conjecture

Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k_K$, absolute ramification degree $e$, and let $\overline\rho_K:G_K\to\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be continuous. For $b_{\overline\tau}\geq1$, set

$$\sigma_{\vec a,\vec b}=\bigotimes_{\overline\tau:k_K\hookrightarrow\overline{\mathbb F}_p}\det^{a_{\overline\tau}}\otimes\operatorname{Sym}^{b_{\overline\tau}-1}k_K^2\otimes_{\overline\tau}\overline{\mathbb F}_p.$$

For each $\overline\tau$, let $\tau^j$ be the embeddings of $K$ lifting it. Local higher-weight conjecture. The following are equivalent: $\overline\rho_K$ has a crystalline lift whose Hodge–Tate weights with respect to $\tau^j$ are $a_{\overline\tau}$ and $a_{\overline\tau}+b_{\overline\tau}$ for $j=1$, and $0$ and $1$ otherwise; and some Jordan–Hölder factor $\sigma'_{\vec a',\vec b'}$ of $\sigma_{\vec a,\vec b}$, with $1\leq b'_{\overline\tau}\leq p$, admits a crystalline lift with the corresponding weights $a'_{\overline\tau}$, $a'_{\overline\tau}+b'_{\overline\tau}$ for $j=1$, and $0$, $1$ otherwise. This conjecture would make the higher-weight formulation equivalent to a purely local assertion about crystalline lifts.