  For $\alpha \ge 1$, $\beta \ge 0$, and a graph $G$, a spanning subgraph $H$
of $G$ is said to be an $(\alpha, \beta)$-spanner if $\dist(u, v, H) \le \alpha
\cdot \dist(u, v, G) + \beta$ holds for any pair of vertices $u$ and $v$. These
type of spanners, called \emph{linear spanners}, generalizes \emph{additive
spanners} and \emph{multiplicative spanners}. Recently, Fomin, Golovach,
Lochet, Misra, Saurabh, and Sharma initiated the study of additive and
multiplicative spanners for directed graphs (IPEC $2020$). In this article, we
continue this line of research and prove that \textsc{Directed Linear Spanner}
parameterized by the number of vertices $n$ admits no polynomial compression of
size $\calO(n^{2 - \epsilon})$ for any $\epsilon > 0$ unless $\NP \subseteq
\coNP/poly$. We show that similar results hold for \textsc{Directed Additive
Spanner} and \textsc{Directed Multiplicative Spanner} problems. This
sparsification lower bound holds even when the input is a directed acyclic
graph and $\alpha, \beta$ are \emph{any} computable functions of the distance
being approximated.
