The local ghost conjecture for primitive augmented modules

Let $\bar\rho=\begin{pmatrix}\omega_1^{a+b+1}&*\\0&\omega_1^b\end{pmatrix}:\operatorname{I}_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb{F})$ be a character with $a\in\{1,\dots,p-4\}$ and $b\in\{0,\dots,p-2\}$. Let $\widetilde{\mathrm{H}}$ be a primitive $\mathcal{O}\llbracket K_p\rrbracket$-projective augmented module of type $\bar\rho$, and let $\varepsilon$ be a character of $\Delta^2$ relevant to $\bar\rho$. Define the characteristic power series $C^{(\varepsilon)}(w,t)$ of the $U_p$-action and the ghost series $G^{(\varepsilon)}(w,t)$ for $\widetilde{\mathrm{H}}$ as in the paper. Local ghost conjecture. The ghost series $G^{(\varepsilon)}(w,t)$ depends only on $\bar\rho$ and $\varepsilon$, and for every $w_\star\in\mathfrak{m}_{\mathbb{C}_p}$ one has

$$\operatorname{NP}(G^{(\varepsilon)}(w_\star,-))=\operatorname{NP}(C^{(\varepsilon)}(w_\star,-)).$$

Part (1) is proved in the paper, while part (2), identifying the Newton polygons of the ghost and characteristic power series, is the remaining conjectural assertion.