Breuil's Ext conjecture for de Rham non-trianguline representations of $\operatorname{GL}_2(\mathbb Q_p)$

Let $D$ be a de Rham $(\varphi,\Gamma)$-module of rank $2$ over $\mathcal R_E$ with Hodge–Tate weights $(0,k)$, and let $\Delta$ be the associated $p$-adic differential equation. Write $\pi_c(\Delta,k)$ and $\pi_{\operatorname{alg}}(\Delta,k)$ for the corresponding locally analytic representations, and let $\mathcal L(D)$ be the $E$-line in $D_{\operatorname{dR}}(\Delta)$ determined by the Hodge filtration. The representation $\pi(D)$ determines an extension class

$$[\pi(D)]\in \operatorname{Ext}^1_{\operatorname{GL}_2(\mathbb Q_p)}\bigl(\pi_c(\Delta,k),\pi_{\operatorname{alg}}(\Delta,k)\bigr).$$

Breuil's conjecture. There is a natural $E$-linear bijection

$$\operatorname{Ext}^1_{\operatorname{GL}_2(\mathbb Q_p)}\bigl(\pi_c(\Delta,k),\pi_{\operatorname{alg}}(\Delta,k)\bigr)\xrightarrow{\sim}D_{\operatorname{dR}}(\Delta)$$

such that, for every such $D$ whose associated $p$-adic differential equation is isomorphic to $\Delta$, the map sends the $E$-line $E[\pi(D)]$ to $\mathcal L(D)$. This conjecture predicts that the extension class of the locally analytic representation records precisely the filtration parameter omitted by the differential equation and the Hodge–Tate weight.