Let $G=(V,A)$ be a directed acyclic graph, for which the underlying undirected graph is chordal (so that every induced cycle in the underlying undirected graph is a triangle).
It is known that in a chordal graph the number of maximum cliques is linearly bounded in the number $|V|$ of vertices.
Let $G'=(V,A')$ be the transitive closure of $G$, so that for every directed path $p= (v_i,\ldots,v_j)$ from $v_i$ to $v_j$ in $G$, there exists a directed edge $(v_i,v_j)$ in $A'$.
Question: Is there any polynomial/linear bound on the number of maximum cliques in the undirected graph that underlies $G'$?
Consider the following graph, with edges oriented from left to right :
This graph is chordal.
The transitive closure of this graph is the complete graph without every vertical edges.
A clique in the transitive closure is any subset of the vertices such that there isn't two vertices with the same $x$ coordinate.
A maximal clique is thus a maximal set of such vertices, so one for each different $x$ coordinate.
There is an exponential number of those : $2^{\frac {n-1} 3}$.
This construction isn't optimal, but I guess this lower bound gives the idea.