Let $A$ be a representation-finite quiver algebra. In this case $A$ is simply connected if and only if its first Hochschild cohomology vanishes by a result of Buchweitz and Liu. $A$ is called strongly simply connected in case every convex subcategory of $A$ is simply connected.
The importance of this notion is for example shown in the following result of Bongartz which gives a strong generalisation of Gabriel's theorem:
Theorem: In case $A$ is representation-finite and strongly simply connected, there is a bijection between the indecomposable $A$-modules and the set of positive roots of the Tits form of $A$ (sending a dimension vector to a root).
Question 1: Is there an easy example showing that the theorem is false when we replace "strongly simply connected" by "simply connected"?
Question 2: Is there a (quick) test with the computer (using QPA) to check whether a given representation-finite quiver algebra is strongly simply connected? The direct way would probably to check whether $HH^1(eAe)=0$ for any basic idempotent $e$, which is possible but takes too long with the computer for most algebras.
(Question 3 had a negative answer so I removed it)
