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This paper is a contribution to the study of relative holonomic$\mathcal{D}$-modules. Contrary to the absolute case, the standard$t$-structure on holonomic $\mathcal{D}$-modules is not preserved by dualityand hence the solution functor is no longer $t$-exact with respect to thecanonical, resp. middle-perverse, $t$-structures. We provide an explicitdescription of these dual $t$-structures. When the parameter space is1-dimensional, we use this description to prove that the solution functor aswell as the relative Riemann-Hilbert functor are $t$-exact with respect to thedual $t$-structure and to the middle-perverse one while the de Rham functor is$t$-exact for the canonical, resp. middle-perverse, $t$-structures and theirduals.

$t$-Structures for Relative $\mathcal{D}$-Modules and $t$-Exactness of the de Rham Functor