Here we show that every cluster variable in a simply laced doubly extended cluster algebra occurs on the tail of a Tpqr quiver. There are four cases to consider: D4,E6,E7,E8. For each case we showed that there is a mutation path freezing each node in every quiver isomorphism class to a quiver with a double edge. This is sufficient as freezing the nodes on the double edge in any such quiver in these cases results in a product of finite cluster algebras of type An. It is well known every cluster variable in An appears on a sources/sink orientation of quiver isomorphic to a path.
The following files are formatted similarly to the output of Kellers Mutation App. Each case starts with //Matrix. The next line contains the dimensions of the adjacency matrix. This is followed by the adjacency matrix for the quiver isomorphism class being considered. The nodes of the quiver are labeled 1 up to n. After this is the tag //Paths To Double Edge. The next lines are of the form i:path. This represents a path avoiding node i that ends in a quiver with a double edge so node i isn't on the double edge. Each path is a comma separated list of locations to mutate at. An empty path implies that the quiver already has a double edge and node i is not on it.
Note that for the E8 Doubly extended we only found paths from each undirected quiver isomorphism class. This suffices as there are mutation paths between directed isomorphism classes freezing any node.