Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in those papers are acyclic, so my question is
Can a directed acyclic graph be reconstructed from its deck of vertex-deleted subgraphs?
One has to assume the graph has at least $5$ vertices to avoid certain small cases. (Edit: For $4$ vertices, see Julian's example below.) Acyclic tournaments are reconstructible according to the references.
The question has an equivalent reformulation in representation theory:
Let $Q$ be a directed acyclic graph as above, and let $k$ be an algebraically closed field. Can the path algebra $\Lambda=kQ$ be reconstructed from its deck of vertex-deleted quotients $\Lambda/\Lambda e \Lambda$?