Jeker–Herzig–Breuil conjecture on extensions of parabolically induced representations

Let $F/\mathbb{Q}_p$ and $k/\mathbb{F}_p$ be finite extensions. Let $\mathbf{G}$ be a split connected reductive $F$-group with connected centre and simply connected derived subgroup, and let $\mathbf{P}=\mathbf{L}\mathbf{N}$ and $\mathbf{P}'=\mathbf{L}'\mathbf{N}'$ be standard parabolic subgroups. Let $\sigma$ and $\sigma'$ be supersingular representations of $L$ and $L'$ over $k$, respectively, and suppose that $\operatorname{Ind}_{P^-}^G\sigma$ and $\operatorname{Ind}_{P'^-}^G\sigma'$ are irreducible or that $p\ne2$. Jeker–Herzig–Breuil conjecture. The following assertions hold: (i) if $\mathbf{P}'\not\subseteq\mathbf{P}$ and $\mathbf{P}\not\subseteq\mathbf{P}'$, then

$$\operatorname{Ext}_G^1\left(\operatorname{Ind}_{P'^-}^G\sigma',\operatorname{Ind}_{P^-}^G\sigma\right)=0;$$

(ii) if $F=\mathbb{Q}_p$, $\mathbf{P}'=\mathbf{P}$, and $\sigma'\cong\sigma^\alpha\otimes(\omega^{-1}\circ\alpha)\not\cong\sigma$ for some $\alpha\in\Delta_{\mathbf{L}}^\perp$, then

$$\dim_k\operatorname{Ext}_G^1\left(\operatorname{Ind}_{P^-}^G\sigma',\operatorname{Ind}_{P^-}^G\sigma\right)=1;$$

(iii) otherwise, if $\mathbf{P}'\subseteq\mathbf{P}$, then parabolic induction induces a $k$-linear isomorphism

$$\operatorname{Ext}_L^1\left(\operatorname{Ind}_{L\cap P'^-}^L\sigma',\sigma\right)\overset{\sim}{\longrightarrow}\operatorname{Ext}_G^1\left(\operatorname{Ind}_{P'^-}^G\sigma',\operatorname{Ind}_{P^-}^G\sigma\right);$$

(iv) otherwise, if $\mathbf{P}\subseteq\mathbf{P}'$, then parabolic induction induces a $k$-linear isomorphism

$$\operatorname{Ext}_{L'}^1\left(\sigma',\operatorname{Ind}_{L'\cap P^-}^{L'}\sigma\right)\overset{\sim}{\longrightarrow}\operatorname{Ext}_G^1\left(\operatorname{Ind}_{P'^-}^G\sigma',\operatorname{Ind}_{P^-}^G\sigma\right).$$

This predicts a complete description of first extensions between irreducible parabolically induced mod-$p$ representations, including vanishing, one-dimensional exceptional extensions, and extensions inherited from Levi subgroups. The conjecture was formulated in the cited prior work; no resolution status is supplied here.