Let algebras always be finite dimensional connected non-semisimple quiver algebras. Say an algebra $A$ has property * in case $eAe$ is a Nakayama algebra, when $eA$ denotes the basic version of the direct sum of all indecomposable projective-injective modules. Is there a nice other characterisation of algebras having property *? This class of algebras seems to contain several large classes such as:

-monomial algebras

-Algebras with no projective-injective module.

Are there other large known classes of algebras having property *?