Injectivity and projectivity of adelic blocks

Let $F$ be a global field, let $\mathbb{A}$ denote its ring of adèles, and let $\mathsf{LCA}_{F}$ be the exact category of locally compact $F$-vector spaces. An adelic block is an object $G$ of $\mathsf{LCA}_{F}$ of the type considered in the paper.

Adelic-block conjecture. Every adelic block $G\in\mathsf{LCA}_{F}$ is both an injective and projective object.

The claim concerns the injective–projective symmetry suggested by Pontryagin self-duality in the locally compact setting; the paper states that it could not settle this question.