Let $A$ be a quiver algebra with an acyclic quiver and primitive idempotents $e_i$. The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $dim(e_i A e_j)$ and the Coxeter matrix $\phi_A$ of $A$ is defined as $\phi_A=-C_A^{-1} C_A^T$. The Coxeter polynomial of $A$ is defined as characteristic polynomial of $\phi_A$. The Coxeter polynomial is a derived invariant and thus derived equivalent algebras share the same Coxeter polynomial.

Is the following true:

A is derived equivalent to a path algebra $KQ$ of Dynkin type if and only if it has the same Coxeter polynomial as $KQ$?

In case the answer is no, is this true in case $A$ is additionally a representation-finite or a Nakayama algebra with a linear quiver?

I think this is at least true when $A$ is a Nakayama algebra with a linear quiver (corresponding to a Dyck path) and there is computational way using trivial extensions and representation-finiteness of those trivial extensions to test it, but it gets very ugly when $Q$ is of type $D_n$ ($E_n$ can be done with the computer and indeed, for $E_6$ it is true. It is also true for Dynkin type $A_n$ and $n \leq 8$ and Dynkin type $D_n$ for $n \leq 6$).