Relative regular Riemann-Hilbert correspondence

On the product of a complex manifold $X$ by a complex curve $S$ considered as a parameter space, we show a Riemann-Hilbert correspondence between regular holonomic relative $\mathcal D$-modules (resp. complexes) on the one hand and relative perverse complexes (resp. $S$-$\mathbb{C}$-constructible complexes) on the other hand.
Comment: 22 pages. This article improves and supersedes arXiv:1811.07151. V2: 28 pages, various corrections, new Th.2, improvement of the presentation. V3: final version to appear in the Proceedings of the London Mathematical Society. V4 Version post-publication, corrects an error in the proof of Prop.3.3 of the published version, Sections 3.2-3.4 modified