Extending meromorphic connections to coadmissible D-cap-modules

We investigate when a meromorphic connection on a smooth rigid analytic variety $X$ gives rise to a coadmissible $\mathcal{D}_X$-cap-module, and show that this is always the case when the roots of the corresponding $b$-functions are all of positive type. On the other hand, we also give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.
Comment: 21 pages. To appear in Crelle's Journal