The conjecture identifying the boundary sheaf of the Drinfeld half-plane covering

Let $c1$ be the representation occurring in the definition of the sheaf $c4_n^{-1}(V-B_i)^{c1}$, and let $a3_n$ be the corresponding covering of the Drinfeld half-plane. For an open subset $U$ of $bb P^1(bb Q_p)$, let $calF_{c0}(U)$ denote the inductive-limit sheaf of functions on the inverse images of the corresponding regions outside larger and larger balls in the Bruhat--Tits tree. Boundary-sheaf conjecture. The sheaf $calF_{c0}$ on $bb P^1(bb Q_p)$ is the sheaf $Umapsto tN_{\rm rig}(c0)\boxtimes U$ of Colmez, and the exact sequence of $G$-representations

$$0\to\Pi(c0,0)^*\to tN_{\rm rig}(c0)\boxtimesbb P^1(bb Q_p)\to\Pi(c0,2)\to0$$

identifies with the exact sequence induced by the functions sequence

$$0\to\mathcal O(a3_n)^{c1}\to\mathcal F_{c0}(bb P^1(bb Q_p))\to(\Omega^1(a3_n)^{c1})^*\to0.$$

This conjecture is presented as a natural extension of the Breuil--Strauch conjecture and would identify the boundary sections of the Drinfeld covering with Colmez's sheaf; its resolution is not indicated in the supplied text.