Is there a counterexample to the following:

Let A be a symmetric algebra and W the direct sum of all indecomposable projective A-modules P such that soc(P) is periodic. Then EndA(W) is a periodic algebra.

I cant back this up with many examples and I would think that it is false.

Here a non-trivial example: Let $C=A \otimes_K B$ be the tensor product of the two hereditary Nakayama algebras $A$ and $B$ with 3 simple modules. Let $G=C/CeC$ when $eC$ is the minimal faithful projective-injective $C$-module. Let $T$ be the trivial extension of $G$. 4 out of 8 simple modules of $T$ are periodic and the resulting algebra $End_T(W)$ as above is periodic.

As continuation of the previous thread Example to periodic symmetric algebras , I have the following question:

Is there a counterexample to the following:

Let A be a symmetric algebra and W the direct sum of all indecomposable projective A-modules P such that soc(P) is periodic. Then EndA(W) is a periodic algebra.

I cant back this up with many examples and I would think that it is false.

Here a non-trivial example: Let $C=A \otimes_K B$ be the tensor product of the two hereditary Nakayama algebras $A$ and $B$ with 3 simple modules. Let $G=C/CeC$ when $eC$ is the minimal faithful projective-injective $C$-module. Let $T$ be the trivial extension of $G$. 4 out of 8 simple modules of $T$ are periodic and the resulting algebra $End_T(W)$ as above is periodic.