Relative strongly regular holonomic $\mathcal{D}$-modules and the Riemann-Hilbert correspondence

We introduce the notion of strong regular holonomic ${$\mathcal{D}$}_{{X\times S}/S}$-moduleandweprovethatthefunctor${$\mathrm{R}\mathrm{H}$}^S$ introduced by T. Monteiro Fernandes and C. Sabbah in [14] takes image in ${\mathsf{D}}_{\mathrm{s}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{l}}^{\mathrm{b}}({\mathcal{D}}_{X\times S\mathrm{/}S})$ (complexes of ${\mathcal{D}}_{X\times S\mathrm{/}S}$-module whose cohomologies are strongly regular). We prove that for $\dim X=\dim S=1$ the functor solution functor ${}^{\mathrm{p}}\mathrm{S}\mathrm{o}\mathrm{l}$ restricted to ${\mathsf{D}}_{\mathrm{s}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{l}}^{\mathrm{b}}({\mathcal{D}}_{X\times S\mathrm{/}S})$ is an equivalence of categories with quasi-inverse ${\mathrm{R}\mathrm{H}}^S$.