Essential-image conjecture for the generalized Montréal functor

Let $\Delta=\{\alpha_1,\dots,\alpha_{n-1}\}$, and let $V$ be an object of $\operatorname{Rep}^{\operatorname{ord}}_h(G_{\mathbb{Q}_p,\Delta}\times\mathbb{Q}_p^\times)$. For each $\alpha=\alpha_j\in\Delta$, let

$$0=\operatorname{soc}_0^{\overline{\alpha}}(V)\lneq\operatorname{soc}_1^{\overline{\alpha}}(V)\lneq\dots\lneq\operatorname{soc}_{r_\alpha}^{\overline{\alpha}}(V)=V$$

be the socle filtration as a representation of $G_{\mathbb{Q}_p,\Delta\setminus\{\alpha\}}\times\mathbb{Q}_p^\times$, and write $\mathrm{gr}^{\overline{\alpha}}_i(V)(\eta_{\overline{\alpha}})$ for the $\eta_{\overline{\alpha}}$-isotypic component of its $i$th graded piece, where $\eta_{\overline{\alpha}}$ is a character of $G_{\mathbb{Q}_p,\Delta\setminus\{\alpha\}}\times\mathbb{Q}_p^\times$. Let $\widetilde{\eta_{\overline{\alpha}}}^{(k)}$ denote the character associated with the restriction to the $k$th copy of $G_{\mathbb{Q}_p}$ for $k\ne j$, to the central $\mathbb{Q}_p^\times$ for $k=n$, and let $\widetilde{\eta_{\overline{\alpha}}}^{(0)}$ be the trivial character. Essential-image conjecture. The object $V$ is in the essential image of $\mathbb{V}^\vee\circ D^\vee_\Delta$ if and only if, for every $\alpha\in\Delta$, $i\in\{1,\dots,r_\alpha\}$, and character $\eta_{\overline{\alpha}}\colon G_{\mathbb{Q}_p,\Delta\setminus\{\alpha\}}\times\mathbb{Q}_p^\times\to\mathbb{F}_q^\times$, the representation $\mathrm{gr}^{\overline{\alpha}}_i(V)(\eta_{\overline{\alpha}})$ is in the image of Colmez's Montréal functor evaluated at a representation $\pi$ of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\mathbb{F}_q$ with central character

$$\frac{\widetilde{\eta_{\overline{\alpha}}}^{(j+1)}}{\widetilde{\eta_{\overline{\alpha}}}^{(j-1)}}.$$

This conjecture characterizes the essential image of the generalized Montréal functor using the socle filtrations of ordinary mod $p$ representations; its resolution would give a precise criterion for recognizing objects arising from $\mathbb{V}^\vee\circ D^\vee_\Delta$.