  A well-known conjecture of Tuza asserts that if a graph has at most $t$
pairwise edge-disjoint triangles, then it can be made triangle-free by removing
at most $2t$ edges. If true, the factor 2 would be best possible. In the
directed setting, also asked by Tuza, the analogous statement has recently been
proven, however, the factor 2 is not optimal. In this paper, we show that if an
$n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed
triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all
directed triangles. We complement our result by presenting two constructions of
large directed graphs with $t\in\Omega(n^2)$ whose smallest such set has
$1.5t-o(n^2)$ arcs.
