Bergdall–Pollack slope-invariant conjecture for ghost series under direct sum

Let $p\geq 7$ be prime, let $N$ be coprime to $p$, and let $\mathrm{K}_p=\operatorname{GL}_2(\mathbb{Z}_p)$. Let $\mathcal{O}_{\mathbb{C}_p}$ be the ring of integers of $\mathbb{C}_p$, with maximal ideal $\mathbf{m}_{\mathbb{C}_p}$. Given two $\mathcal{O}\llbracket \mathrm{K}_p\rrbracket$-projective augmented modules $\widetilde{\mathrm{H}}_1$ and $\widetilde{\mathrm{H}}_2$ of related type, let $G_1(w,-)$, $G_2(w,-)$, and $G_3(w,-)$ denote, respectively, the corresponding ghost series of $\widetilde{\mathrm{H}}_1$, $\widetilde{\mathrm{H}}_2$, and $\widetilde{\mathrm{H}}_1\oplus\widetilde{\mathrm{H}}_2$. Bergdall–Pollack's slope-invariant conjecture. The Newton polygons of $(G_1\cdot G_2)(w_\star,-)$ and $G_3(w_\star,-)$ should be the same for every $w_\star\in\mathbf{m}_{\mathbb{C}_p}$. This conjecture asserts that the slope data of the ghost series is invariant under direct sum of related-type augmented modules; it is a reformulation of a question posed after the local ghost conjecture. The local ghost conjecture itself is known under a genericity hypothesis, but the direct-sum assertion is not resolved in the supplied text.