  (see paper for full abstract)
  We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by
the number $k$ of terminal pairs, even when the input graph is a planar
directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03,
SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that
there is no $f(k)\cdot n^{o(k)}$ algorithm for Edge-Disjoint Paths on planar
DAGs, where $k$ is the number of terminal pairs, $n$ is the number of vertices
and $f$ is any computable function. Our hardness holds even if both the maximum
in-degree and maximum out-degree of the graph are at most $2$.
  Our result shows that the $n^{O(k)}$ algorithm of Fortune et al. (TCS '80)
for Edge-Disjoint Paths on DAGs is asymptotically tight, even if we add an
extra restriction of planarity. As a special case of our result, we obtain that
Edge-Disjoint Paths on planar directed graphs is W[1]-hard parameterized by the
number $k$ of terminal pairs. This answers a question of Cygan et al. (FOCS
'13) and Schrijver (pp. 417-444, Building Bridges II, '19), and completes the
landscape of the parameterized complexity status of edge and vertex versions of
the Disjoint Paths problem on planar directed and planar undirected graphs.
