The Schneider–Ollivier surjectivity conjecture for the canonical morphism

Let $G$ be a reductive $p$-adic Lie group, let $\mathbb{X}$ be the associated coefficient system, and let $\mathcal{H}$ be its Hecke algebra. The canonical morphism is

$$\mathbb{X}^*\to\mathcal{H}.$$

Schneider–Ollivier's surjectivity conjecture. The canonical morphism $\mathbb{X}^*\to\mathcal{H}$ is a surjection.

This conjecture was denoted by (sur) by Schneider and Ollivier and is described as the largest obstacle to understanding the torsionfree category of $\mathcal{H}$-modules in higher rank. Its resolution status is not specified in the source.