  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 duality
and hence the solution functor is no longer $t$-exact with respect to the
canonical, resp. middle-perverse, $t$-structures. We provide an explicit
description of these dual $t$-structures. When the parameter space is
1-dimensional, we use this description to prove that the solution functor as
well as the relative Riemann-Hilbert functor are $t$-exact with respect to the
dual $t$-structure and to the middle-perverse one while the de Rham functor is
$t$-exact for the canonical, resp. middle-perverse, $t$-structures and their
duals.
