Ext-finite modules for weakly symmetric algebras with radical cube zero

Assume $A$ is weakly symmetric, indecomposable, with radical cube zero and radical square non-zero. We show that such algebra of wild representation type does not have a non-projective module $M$ whose ext algebra is finite-dimensional. This gives a complete classification weakly symmetric indecomposable algebras which have a non-projective module whose ext algebra is finite-dimensional.