Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor

Let $\rho$ be an $n$-dimensional representation of ${\mathcal{G}}_{{\mathbb{Q}}_p}$ over $\overline{{\mathbb{F}}_p}$. When $\rho$ is generic and a good conjugate, the article “Conjectures and results on modular representations of ${\mathrm{G}\mathrm{L}}_n(K)$ for a $p$-adic field $K$”, by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of compatibility with $\rho$ for an admissible representation of ${\mathrm{G}\mathrm{L}}_n({\mathbb{Q}}_p)$. In loc. cit., the five authors also question whether one could recover a representation of ${\mathcal{G}}_{{\mathbb{Q}}_p}^{n-1}$, called ${\bar{L}}^{⊠}(\rho )$ and constructed from $\rho$, from some $\mathrm{\Pi }$ compatible with $\rho$ by using Zábrádi’s functor ${\mathbf{V}}_{\mathrm{\Delta }}$. We give a range of results, for an arbitrary $\mathrm{\Pi }$ verifying some “weak” compatibilities with $\rho$, about how badly ${\mathbf{V}}_{\mathrm{\Delta }}(\mathrm{\Pi })$ behaves. In particular, when $\rho$ is reducible and $n\ge 3$, no $\mathrm{\Pi }$ compatible with $P_{\rho }$ can verify ${\mathbf{V}}_{\mathrm{\Delta }}(\mathrm{\Pi })\simeq {\bar{L}}^{⊠}(\rho )$.